PS

212. (Book Question: 122)If n is a positive integer and the product of all the integers from 1 to n, inclusive, is divisible by 990, what is the least possible value of n ? A. 8 B. 9 C. 10 D. 11 E. 12

For convenience, let N represent the product of all integers from 1 through n. Then, since N is divisible by 990, every prime factor of 990 must also be a factor of N. The prime factorization of 990 is 2 × 32 × 5 × 11, and therefore, 11 must be a factor of N. Then, the least possible value of N with factors of 2, 5, 32, and 11 is 1 × 2 × 3 × . . . × 11, and the least possible value of n is 11.

prime number

不包括1

148-答案

不要忘记了最后一个月的还是很少，要拉通了看

If 0 < x < y, what is the value of (x + y)^2/(x- y)^2? (1) x^2 + y^2 = 3xy (2) xy = 3

1) x^2+y^2=3xy => x^2+y^2-2xy=xy => (x-y)^2=xy So you can replace : (x+y)^2/xy And then just finish the work : (x+y)^2/xy => (x^2+y^2+2xy)/xy => (3xy+2xy)/xy => 5 1 is enough 2) not enough. (x+y)^2/(x-y)^2 => (x^2+y^2+2xy)/(x^2+y^2-2xy) => (x^2+y^2+6)/(x^2+y^2-6) => you can't know the value of x^2 or y^2 Hope it helps.

199-答案

Approach #1: Even integer between 99 and 301 represent evenly spaced set (aka arithmetic progression): 100, 102, 104, ..., 300. Now, the sum of the elements in any evenly spaced set is the mean (average) multiplied by the number of terms. (Check Number Theory chapter of Math Book for more: math-number-theory-88376.html) *Average of the set: (largest+smallest)/2=(300+100)/2=200; # of terms: (largest-smallest)/2+1=(300-100)/2+1=101 (check this: totally-basic-94862.html#p730075);* The sum = 200*101= 20,200. Answer: B. Approach #2: *Using the formula of the sum of the first n positive integers: n(n+1)/2*. 100+102+...+300=2(50+51+..+150). Now, the sum of the integers from 50 to 150, inclusive equals to the sum of the integers from 1 to 150, inclusive minus the sum of the integers from 1 to 49, inclusive. 2(50+51+..+150)=2*(150(150+1)/2-49(49+1)/2)=20,200.

In the xy-plane, the line k passes through the origin and through the point (a,b), where ab does not equal 0. Is b positive? (1) The slope of line k is negative (2) a < b

1. the slope of line k is negative 2. a < b[/quote] (1) only tells that (a,b) must fall in II or IV With (1)(2) together we have a<b so (a,b) must fall in II and b is positive The answer is C

Rates for having a manuscript typed at a certain typing service are $5 per page for the first time a page is typed and $3 per page each time a page is revised. If a certain manuscript has 100 pages, of which 40 were revised only once, 10 were revised twice, and the rest required no revisions, what was the total cost of having the manuscript typed? A. $430 B. $620 C. $650 D. $680 E. $770

100*5 +40*3 +10*2*3 =680 Is D the answer

In the figure above, equilateral triangle ABC is inscribed in the circle. If the length of arc ABC is 24, what is the approximate diameter of the circle?

Arc ABC is 2323 of the circumference (as ABC is equilateral triangle and thus arc AB=arc BC=arc AC, so arc AB+arc BC=arc ABC = 2/3 of circumference) --> 24=c∗2324=c∗23, hence circumference c=24∗32=36=πdc=24∗32=36=πd --> d≈11.5d≈11.5.

214. (Book Question: 98)The sum of all the integers k such that -26 < k < 24 is A. 0 B. -2 C. -25 D. -49 E. -51

Correct Answer: D Selected Answer: B Arithmetic: Operations on integers In the sum of all integers k such that -26 < k < 24, the positive integers from 1 through 23 can be paired with the negative integers from -1 through -23. The sum of these pairs is 0 because a + (-a) = 0 for all integers a. Therefore, the sum of all integers k such that -26 < k < 24 is -25 + (-24) + (23)(0) = -49.

The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers? A. 17 B. 16 C. 15 D. 14 E. 13

I think this question has been discussed earlier also. squares of natural numbers, which are below 75,are 1,4,9,16,25,36,49,64 1+25+49=75 is the only option so numbers are 1,5,7 sum =13

Last year, sales at Company X were 10% greater in February than in January, 15% less in March than in Feb, 20% greater in April than in March, 10% less in May than in April, and 5% greater in June than in May. In which month were sales closest to Jan? a. Feb b. Mar c. Apr d. May e. June

This can be solved using smart numbers: Let January = $100. (Keep in mind we are using "approximate" values, as per the problem). We have: Jan = $100 10% greater in February than in January ---> Feb = 1.1 Jan = $110 15% less in March than in Feb ---> Mar = 0.85 Feb = $93.5 20% greater in April than in March ---> Apr = 1.2 Mar = $112 10% less in May than in April ---> May = 0.9 Apr = $101 (we have a winner) 5% greater in June than in May --->Jun = 1.05 May = $106 Ans D

In the xy plane, at what points does the graph of y=(x+a)(x+b) intersect the x-axis? (1) a + b = -1 (2) The graph intersects the y axis at (0,-6)

X-intercepts of the function f(x)f(x) or in our case the function (graph) y=(x+a)(x+b)y=(x+a)(x+b) is the value(s) of xx for y=0y=0. So basically the question asks to find the roots of quadratic equation (x+a)(x+b)=0(x+a)(x+b)=0. (x+a)(x+b)=0(x+a)(x+b)=0 --> x2+bx+ax+ab=0x2+bx+ax+ab=0 --> x2+(a+b)x+ab=0x2+(a+b)x+ab=0. Statement (1) gives the value of a+ba+b, but we don't know the value of abab to solve the equation. Statement (2) tells us the point of y-intercept, or the value of yy when x=0x=0 --> y=(x+a)(x+b)=(0+a)(0+b)=ab=−6y=(x+a)(x+b)=(0+a)(0+b)=ab=−6. We know the value of abab but we don't know the value of a+ba+b to solve the equation. Together we know the values of both a+ba+b and abab, hence we can solve the quadratic equation, which will be the x-intercepts of the given graph. Answer: C. For more on this topic check Coordinate Geometry chapter of Math Book (link in my signature). Hope it helps.

If x and y are positive integers , is the product xy even (1) 5x - 4y is even (2) 6x + 7y is even

agree with D. 1) 5x - 4y is even -> x must be multiple of 2 5*2k-4y = 2*(any number) even sufficient 2)6x + 7y is even[/ y should multiple of 2 even sufficient _________

When tossed, a certain coin has equal probability of landing on either side. If the coin is tossed 3 times, what is the probability that it will land on the same side each time? A. 1/8 B. 1/4 C. 1/3 D. 3/8 E. 1/2

first toss determines the side (heads or tails). for the sake of this argument, i'll say HEADS. =1 (probability is 1 cause it can go either way) for the second & third toss, there's 1/2 probability that the toss will end up w/ HEADS ..... 1 * 1/2 * 1/2 = 1/4

14-答案

不要被题干弄晕

4. (Book Question: 31) A. 1 B. 4/3 C. 17/5 D. 18/5 E. 4

仔细 Correct Answer: B Selected Answer: C Algebra: Simultaneous equations

119-做法

实验法

gmat模考斜率相乘答案

兩垂直線斜率乘積為-1 The line PO has slope =-1/√3 The line QO has slope = t/s PO and QO is perpendicular so [-1/√3]*t/s = -1 ==> t/s =√3 or √3/1 so s = 1

211-答案

怎么巧妙的拆分最重要

168-答案

Correct Answer: B Selected Answer: E

134-答案

这种整除的题目的解答方法,吧这种思维记住

Of the three-digit positive integers whose three digits are all different and nonzero, how many are odd integers greater than 700? (A) 84 (B) 91 (C) 100 (D) 105 (E) 243

1. It is a 3 digit number > 700 (or between 701-999, inclusive). 2. All digits are different and NON ZERO. 3. The numbers must be ODD --> the last digit can be 1 of 1,3,5,7,9 Based on this, the numbers can be of the following 3 types: Type 1: 7AB Type 2: 8EF Type 3: 9CD For type 1 and type 3, be very careful that the digits must be different. So if it is 7AB, then B can NOT be 7. Similarly for type 3, 9CD, D can NOT be 9. There is no such restriction when you find numbers of type 2 (8EF). Number of combinations for type 1 : 1*7*4 = 28 Number of combinations for type 2: 1*7*5 = 35 Number of combinations for type 3 : 1*7*4 = 28 Thus, total numbers possible = 28+35+28 = 91.

If x is an integer, is the median of the 5 numbers shown greater than the average (arithmetic mean ) of the 5 numbers ? (1) x > 6 (2) x is greater than the median of the 5 numbers.

12不是上线 We have a set: {1, 3, 8, 12, x} Question: is median>mean=x+1+3+8+125=x+245median>mean=x+1+3+8+125=x+245? Note that as we have odd (5) # of terms in the set then the median will be the middle term when arranged in ascending (or descending) order. So, if x≤3x≤3: {1, x, 3, 8, 12} then median=3median=3, if 3<x≤83<x≤8: {1, 3, x, 8, 12} then median=xmedian=x and if x≥8x≥8: {1, 3, 8, x, 12} then median=8median=8. (1) x>6x>6. If x=7x=7 then the median will be 7 as well: {1, 3, 7, 8, 12} and mean will be mean=7+245=6.2mean=7+245=6.2, so median=7>mean=6.2median=7>mean=6.2 and the answer is YES BUT if xx is very large number then the median will be 8: {1, 3, 8, 12, x=very large number} and mean will be more than median (for example if x=26x=26 then mean=26+245=10mean=26+245=10, so median=8<10=meanmedian=8<10=mean) and the answer will be NO. Not sufficient. (2) x is greater than the median of the 5 numbers --> so median=8median=8: now, if x=11x=11 then mean=11+245=7mean=11+245=7, so median=8>7=meanmedian=8>7=mean and the answer is YES. Again it's easy to get answer NO with very large xx. Not sufficient. (1)+(2) Again, x=11 and x=very large number give two diffrent answers to the question. Not sufficeint. Answer: E.

排列组和 C P运算法则 P9/12 C9/12不同算法

1．排列及计算公式 从n个不同元素中，任取m(m≤n)个元素按照一定的顺序排成一列，叫做从n个不同元素中取出m个元素的一个排列；从n个不同元素中取出m(m≤n)个元素的所有排列的个数，叫做从n个不同元素中取出m个元素的排列数，用符号p(n，m)表示. p(n，m)=n(n-1)(n-2)......(n-m+1)=n！/(n-m)！(规定0！=1). P9/12=12!/9! 2．组合及计算公式 从n个不同元素中，任取m(m≤n)个元素并成一组，叫做从n个不同元素中取出m个元素的一个组合；从n个不同元素中取出m(m≤n)个元素的所有组合的个数，叫做从n个不同元素中取出m个元素的组合数.用符号c(n，m)表示. c(n，m)=p(n，m)/m！=n！/((n-m)！*m！)；c(n，m)=c(n，n-m)； C9/12=12!/9!3!

A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q? (1) 1/3 of the people surveyed said that they buy product P but not product Q. (2) 1/2 of the people surveyed said that they buy product Q.

A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q? You can solve this question with Venn diagram, matrix or as shown below. {Total} = {buy P} + {buy Q} - {buy both P and Q} + {buy neither P nor Q}. Question: {buy neither P nor Q} / {Total} = ? Take total to be equal to 6 (as it's a multiple of both 2 and 3) (1) 1/3 of the people surveyed said that they buy product P but not product Q: {buy P} - {buy both P and Q} = 1/3*6 = 2; 6 = {buy P} + {buy Q} - ({buy P} - 2) + {buy neither P nor Q} 4={buy Q} + {buy neither P nor Q}. Not sufficient to get the ratio we need. (2) 1/2 of the people surveyed said that they buy product Q: {buy Q}=1/2*6=3. Not sufficient. (1)+(2) 4={buy Q} + {buy neither P nor Q} and {buy Q} = 3; {buy neither P nor Q} = 1; {buy neither P nor Q}/{Total} = 1/6. Sufficient. Answer: C.

216. (Book Question: 129)A three-digit code for certain locks uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 according to the following constraints. The first digit cannot be 0 or 1, the second digit must be 0 or 1, and the second and third digits cannot both be 0 in the same code. How many different codes are possible? A. 144 B. 152 C. 160 D. 168 E. 176

Correct Answer: B Selected Answer: B Arithmetic: Elementary combinatorics Since the first digit cannot be 0 or 1, there are 8 digits possible for the first digit. Since the second digit must be 0 or 1, there are 2 digits possible for the second digit. If there were no other restrictions, all 10 digits would be possible for the third digit, making the total number of possible codes 8 × 2 × 10 = 160. But, the additional restriction that the second and third digits cannot both be 0 in the same code eliminates the 8 codes 2-0-0, 3-0-0, 4-0-0, 5-0-0, 6-0-0, 7-0-0, 8-0-0, and 9-0-0. Therefore, there are 160 - 8 = 152 possible codes.

108. N=20!+17, it is divisible by which of the following A. None B. I only C. II only D. I and II E. II and III

Correct Answer: C Selected Answer: C Arithmetic: Properties of numbers Because 20! is the product of all integers from 1 through 20, it follows that 20! is divisible by each integer from 1 through 20. In particular, 20! is divisible by each of the integers 15, 17, and 19. Since 20! and 17 are both divisible by 17, their sum is divisible by 17, and hence the correct answer will include II. If n were divisible by 15, then n - 20! would be divisible by 15. But, n - 20! = 17 and 17 is not divisible by 15. Therefore, the correct answer does not include I. If n were divisible by 19, then n - 20! would be divisible by 19. But, n - 20! = 17 and 17 is not divisible by 19. Therefore, the correct answer does not include III.

All boxes in a certain warehouse were arranged in stacks of 12 boxes each, with no boxes left over. After 60 additional boxes arrived and no boxes were removed, all the boxes in the warehouse were arranged in stacks of 14 boxes each, with no boxes left over. How many boxes were in the warehouse before the 60 additional boxes arrived? (1) There were fewer than 110 boxes in the warehouse before the 60 additional arrived. (1) There were fewer than 120 boxes in the warehouse after the 60 additional arrived.

Given: 12x + 60 = 14y y = 12(x + 5)/14 = 6(x + 5)/7 6 is not divisible by 7. So, (x + 5) has to be divisible by 7 --> (x + 5) = 7k x = 7k - 5. Therefore x = 2, 9, 16, 23, .......... St1: There were fewer than 110 boxes in the warehouse before the 60 additional arrived. --> 12x < 110 When x = 2 --> 12x = 24 < 110 When x = 9 --> 12x = 108 < 110 When x = 16 --> 12x = 192 > 110 There are 2 possible values --> 24 or 108 Not Sufficient St2: There were fewer than 120 boxes in the warehouse after the 60 additional arrived. --> 12x + 60 < 120 When x = 2 --> 12x + 60 = 84 < 120 When x = 9 --> 12x + 60 = 168 > 120 There is only one possible value --> 84 Sufficient Answer: B

If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ? (1) The remainder when p + n is divided by 5 is 1. (2) The remainder when p - n is divided by 3 is 1.

If p and n are positive integers and p>n, what is the remainder when p^2 - n^2 is divided by 15? First of all p2−n2=(p+n)(p−n)p2−n2=(p+n)(p−n). (1) The remainder when p + n is divided by 5 is 1. No info about p-n. Not sufficient. (2) The remainder when p - n is divided by 3 is 1. No info about p+n. Not sufficient. (1)+(2) "The remainder when p + n is divided by 5 is 1" can be expressed as p+n=5t+1p+n=5t+1 and "The remainder when p - n is divided by 3 is 1" can be expressed as p−n=3k+1p−n=3k+1. Multiply these two --> (p+n)(p−n)=(5t+1)(3k+1)=15kt+5t+3k+1(p+n)(p−n)=(5t+1)(3k+1)=15kt+5t+3k+1, now first term (15kt) is clearly divisible by 15 (r=0), but we don't know about 5t+3k+1. For example t=1 and k=1, answer r=9 BUT t=7 and k=3, answer r=0. Not sufficient.

If x and y are integers greater than 1, is x a multiple of y? (1) 3y^2+7y=x (2) x^2-x is a multiple of y

If x and y are integers great than 1, is x a multiple of y? (1) 3y2+7y=x3y2+7y=x --> y(3y+7)=xy(3y+7)=x --> as 3y+7=integer3y+7=integer, then y∗integer=xy∗integer=x --> xx is a multiple of yy. Sufficient. (2) x2−xx2−x is a multiple of yy --> x(x−1)x(x−1) is a multiple of yy --> xx can be multiple of yy (x=2x=2 and y=2y=2) OR x−1x−1 can be multiple of yy (x=3x=3 and y=2y=2) or their product can be multiple of yy (x=3x=3 and y=6y=6). Not sufficient. Answer: A. Hope it helps.

A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: Blue, Green, Yellow Or Pink. The store packs the notepads in pacakages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible? A. 6 B. 8 C. 16 D. 24 E. 32

Notepads of the same color = 4 (we have 4 colors). As we have two sizes then total for the same color=4*2=8 Notepads of the different colors = 4C3=4 (we should choose 3 different colors out of 4). As we have two sizes then total for the different color=4*2=8 Total=8+8=16 C43算错了

A certain library assesses fines for overdue books as follows. On the first day that a book is overdue, the total fine is $0.10. For each additional day that the book is overdue, the total fine is either increased by $0.30 or doubled, whichever results in the lesser amount. What is the total for a book on the fourth day it is overdue? A. $0.60 B. $0.70 C. $0.80 D. $0.90 E. $1.00

Notice that, fines are already cumulative: "For each additional day that the book is overdue, the total fine is ..." 1st day fine - 0.1 2nd day fine - 0.1*2 = 0.2 (as doubling gives lower value) 3rd day fine - 0.2*2 = 0.4 (as doubling gives lower value) 4th day fine - 0.4 + 0.3 = 0.7 (as doubling gives higher value we add 0.3 this time) Answer: B.

Stock / Number of shares V ------------ 68 W ---------- 112 X ------------ 56 Y ------------ 94 Z ------------ 45 The table shows the number of shares of each of the 5 stocks owned by Mr Sami. If Mr. Sami was to sell 20 shares of Stock X and buy 24 shares of stock Y, what would be the increase in the range of the numbers of shares of the 5 stocks owned by Mr. Sami? A. 4 B. 6 C. 9 D. 15 E. 20

Original range = (highest #) - (lowest #) = W - Z = 112 - 45 = 67; After selling/buying: X=56-20=36 and Y=94+24=118; New range = (new highest #) - (new lowest #) = Y - X = 118-36 = 82; Difference = New range - Original range = 82 - 67 = 15. Answer: D.

rectangular dimension ratio question

Since the model does not have roof, lets find the surface area without roof of ACTUAL room 16 *10 +2*8*10 + 2*8*16 = 576 sq feet 576 sq feet is represented by 2304 square inches.. so 1 sq feet is represented by 2304/576 = 4 square inches.. The answer has to be in inches per feet.. SO if 1 sq feet is represented by 4 square inches.. 1 feet will be reprsented by 2 inches Answer is in inches per feet = 2 inches /1 feet = 2 inches per feet D

What is the average (arithmetic mean) height of the n people in a certain group? (1) The average height of the n/3 tallest people in the group is 6 feet 2(1/2) inches, and the average height of the rest of the people in the group is 5 feet 10 inches. (2) The sum of the heightsof the n people is 178 feet 9 inches.

Stmt1: Avg of n/3 people= 6feet 2(1/2) inches = n/3 * 6feet 2(1/2) inches Avg of 2n/3 people = 5feet 10 inches = 2n /3*5feet 10 inches If we add both above quantities and divide it by n we will get the avg since n will cancel out. So, sufficient. Stmt2: if sum is given to calculate avg we have to divide it by n. But we dont know the value of n so insufficient. Hence A is the answer

For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6? A. 0 B. 1 C. 2 D. 3 E. 4

Top Member of the Month Re: 2 digit positive integers with length 6 [#permalink] New post 29 Jan 2012, 17:21 13 This post received KUDOS Expert's post 22 This post was BOOKMARKED enigma123 wrote: For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6? A. 0 B. 1 C. 2 D. 3 E. 4 I need to understand the concept behind solving this question please. Basically the length of the integer is the sum of the powers of its prime factors. Length of six means that the sum of the powers of primes of the two-digit integer must be 6. First we can conclude that 5 can not be a factor of this integer as the smallest integer with the length of six that has 5 as prime factor is 2^5*5=160 (length=5+1=6), not a two-digit integer. The above means that the primes of the two-digit integers we are looking for can be only 2 and/or 3. n=2p∗3qn=2p∗3q, p+q=6p+q=6. Let's start with the highest value of pp: n=26∗30=64n=26∗30=64 (length=6+0=6); n=25∗31=96n=25∗31=96 (length=5+1=6); n=24∗32=144n=24∗32=144 (length=4+2=6) not good as 144 is a three digit integer. Answer: C.

The main ingredient in a certain prescription drug capsule cost $500 per kilogram. If each capsule contains 600 milligrams of ingredient, what is the cost of the ingredient in a capsule? (1 kilogram = 10^6 milligrams)

kilograms of substance in one capsule = 600/10^6 Cost of substance in one capsule = 500* (600/10^6)

130. (Book Question: 19)Thabo owns exactly 140 books, and each book is either paperback fiction, paperback nonfiction, or hardcover nonfiction. If he owns 20 more paperback nonfiction books than hardcover nonfiction books, and twice as many paperback fiction books as paperback nonfiction books, how many hardcover nonfiction books does Thabo own? A. 10 B. 20 C. 30 D. 40 E. 50

仔细要

Integer 概念

整数就是没有小数位都是零的数 ，即能被1整除的数（如-1,-2,0,1,......）。

10. (Book Question: 132)Last year 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy. Last year no members of the club traveled to both England and France, 6 members traveled to both England and Italy, and 11 members traveled to both France and Italy. How many members of the club traveled to at least one of these three countries last year? A. 52 B. 67 C. 71 D. 73 E. 79

既然说了其中一个是0，三个交叉项也肯定是0

53. (Book Question: 155)At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is 60 cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 56 cents. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52 cents? A. 1 B. 2 C. 3 D. 4 E. 5

看下方程解怎么解答

94-答案

看清楚单位

113. (Book Question: 71)If the range of the six numbers 4, 3, 14, 7, 10, and x is 12, what is the difference between the greatest possible value of x and the least possible value of x ? A. 0 B. 2 C. 12 D. 13 E. 15

看清楚题目说的是什么，什么是所谓的range Correct Answer: D Selected Answer: Not Any Arithmetic: Statistics The range of the six numbers 3, 4, 7, 10, 14, and x is 12. If x were neither the greatest nor the least of the six numbers, then the greatest and least of the six numbers would be 14 and 3. But, this cannot be possible because the range of the six numbers would be 14 - 3 = 11 and not 12 as stated. Therefore, x must be either the greatest or the least of the six numbers. If x is the greatest of the six numbers, then 3 is the least, and x - 3 = 12. It follows that x = 15. On the other hand, if x is the least of the six numbers, then 14 is the greatest, and 14 - x = 12. It follows that x = 2. Thus, there are only two possible values of x, namely 15 and 2, and so the difference between the greatest and least possible values of x is 15 - 2 = 13.

123. (Book Question: 169)A researcher plans to identify each participant in a certain medical experiment with a code consisting of either a single letter or a pair of distinct letters written in alphabetical order. What is the least number of letters that can be used if there are 12 participants, and each participant is to receive a different code? A. 4 B. 5 C. 6 D. 7 E. 8

读题仔细读 Correct Answer: B Selected Answer: D Arithmetic: Elementary combinatorics None of the essential aspects of the problem is affected if the letters are restricted to be the first n letters of the alphabet, for various positive integers n. With the 3 letters a, b, and c, there are 6 codes: a, b, c, ab, ac, and bc. With the 4 letters a, b, c, and d, there are 10 codes: a, b, c, d, ab, ac, ad, bc, bd, and cd. Clearly, more than 12 codes are possible with 5 or more letters, so the least number of letters that can be used is 5.

29. (Book Question: 206)Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope? A. 82 B. 118 C. 120 D. 134 E. 152

这个题看看答案是怎么想的 D