Combinations and Probability

In a plane there are 8 points, no 3 of which are collinear. How many lines do the points determine?

-Method 1: 7+6+5+4+3+2+1=28 -Method 2: Since no 3 points are collinear, every pair of points determines a distinct line. There are nCr(8,2)=28 such lines.

After polling a class of 20 music students by a show of hands, you find that 8 students play guitar and 9 play piano. Based on this, what is the MINIMUM number of students who play both piano and guitar?

0 (maximum would be 8) <-- logic

nCr(n,0) =

1 for any n

Suppose you have 5 shirts, 4 pairs of pants, and 9 ties. How many outfits can be made consisting of a shirt, a pair of pants, and a tie?

1. For each of the 5 shirts, you can wear 4 pairs of pants, so there are 5*4=20 shirt-pants combinations 2. For each 20 shirt-pants combinations, there are 9 ties, so 20*9=180 shirts-pants-tie combinations

In how many distinguishable ways can the 7 letters in the word MINIMUM be arranged, if all the letters are used each time?

1. The word MINIMUM contains 7 letters, which can be permuted 7! ways. The 3 M's can be permuted 3! ways, and the 2 I's can be permuted in 2! ways, so only 1/(3!2!) = 1/12 permutations look different from each other 2. There are 7!/12 = 420 distinguishable ways the letters can be arranged

M&M plain candies come in 6 colors. Assume there are at least 3 of each color in a bag. If you pick three candies from that bag, how many color possibilities are there?

1. There are 6 choices of color for each of the 3 candies selected. 2. Therefore, there are 6x6x6 = 216 color possibilities together

How many 4-digit personal ID numbers are possible if no digit can be used twice and no number can begin with 0?

1. There are 9 digits that could go in the first position. For each of those, there are 9 that could go in the second position; 8 that could go in the third position, and 7 that could go in the fourth position 2. There are 9 x 9 x 8 x 7 = 4536 four-digit numbers

A basketball team has 5 centers, 9 guards, and 13 forwards. Of these, 1 center, 2 guards, and 2 forwards start a game. How many possible starting teams can a coach put on the floor?

1. There are nCr(5,1)=5 ways of choosing the one center, nCr(9,2)=36 ways of choosing the two guards, and nCr(13,2)=78 ways of choosing the 2 forwards. 2. Therefore, there are 5x36x78=14,040 possible starting teams.

A salad bar has 7 ingredients, excluding the dressing. How many different salads are possible where 2 salads are different if they don't include identical ingredients?

1. You can either include or exclude each of the 7 ingredients in your salad, which means there are 2 choices for each ingredient. 2. According to the multiplication rule, there are 2^7 = 128 ways of making these yes-no choices

permutations and combinations on TI-nspire

1. menu 2. 5: probability 3. permutation/combination

License plates on cars in a certain state consist of 3 letters taken from the 26 letters, A through Z, followed by 3 digits taken from the 10 digits, 0 though 9. Which of the following expressions gives the number of distinct license plates that are possible given that repetition of both letters and digits is allowed?

26 ways to choose 3 letters: (26)(26)(26) 10 ways to choose 3 digits: (10)(10)(10) Number of license plates possible: (26^3)*(10^3)

P (A|B)

Probability of A occurring, given that B has occurred —> conditional probability -P(A)*P(B) -P(A and B) / P(B)

P (A ∩ B)

Probability of BOTH A and B occurring: (A ∩ B) is the conjunction of A and B -P(A)*P(B)

P (A ∪ B)

Probability of EITHER A or B occurring: (A ∪ B) —> also known as the disjunction of A and B -P(A)+P(B) -P(A)+P(B)-P(A ∩ B)

Three cards are drawn from an ordinary deck of 52 cards. Each card is replaced in the deck before the next card is drawn. What is the probability that at least one of the cards will be a spade? (There are 13 spades in a deck.)

Since the drawn cards are replaced, the draws are independent. The probability that NONE of the cards was a spade = (39/52)*(39/52)*(39/52)=27/64 -Probability that 1 was a spade: 1-(27/64)=37/64

A coin is tossed three times. Let A = {3 heads occur} and B = {at least one head occurs}. What is P(A ∪ B)?

The only situation when NEITHER of these sets is satisfied occurs when 3 tails appear. -P(A ∪ B) = 7/8

If a coin is flipped and one die is thrown, what is the probability of getting a head or a 4?

The probability of getting NEITHER a head nor a 4 is (1/2)*(5/6)=(5/12) -The probability of getting either is 1-(5/12)=7/12

If the probability that the Giants will win the NFC is p and if the probability that the Raiders will win the AFC is q, what is the probability that only one of these teams will win its respective championship?

The probability that both teams will win is pq. The probability that both will lose is (1-p)(1-q). The probability that only one will win is 1-[pq+(1-p)(1-q)]=p+q-2pq

With the throw of two dice, what is the probability that the sum will be a prime number?

There is 1 way to get a 2, and there are 2 ways to get a 3, 4 ways to get a 5, 6 ways to get a 7, 2 ways to get an 11 -Out of 36 elements in the sample space, 15 successes are possible -P(prime)=(15/16)=(5/12)

25% of a group of unrelated students are only children. The students are asked one at a time whether they are only children. What is the probability that the 5th student asked is the first only child?

Whether or not students in the group have siblings are independent events. The probability that each of the first 4 is not an only child is (0.75)^4. The probability that the 5th student is an only child is 0.25, so the probability of seeing the first 4 children with siblings and the 5th an only child is (0.75)^4(.25)=.08

Fundamental Counting Principle

if an event A can happen in N ways, and another, independent event B can happen in M ways, then both events together can happen in N x M ways.

The student council at Central High has 20 members. They want to select a committee of 3 to work with the school board. How many committees are possible?

nCr(20,3) = 1140

A hotel has 5 single rooms available, for which 6 men and 3 women apply. What is the probability that the rooms will be rented to 3 men and 2 women?

nCr(6,3)=20 is the number of ways 3 men can be selected; nCr(3,2)=3 is the number of ways 2 women can be selected, nCr(9,5)=126 is the TOTAL number of ways people can be selected to fill 5 rooms -P(3 men, 2 women) = (20*3)/126 = 10/21

The math team at East High as 20 members. They want to choose a president, vice president, and treasurer. In how many ways can this be done?

nPr(20,3) = 6480

combination

order does not matter (something CAN be used more than once, creating groups) -the number of ways r objects can be chosen from n -nCr(# of total choices, # groups) OR nCr(#of choices per sample, total # in sample)

permuation

order matters (something CANNOT be used more than once, creating lists) -the number of ways (r) you could choose a first, second, third... etc. out of n selections -nPr(# of total choices, # of positions)

multiplication rule

to determine the probability, we multiply the probability of one event by the probability of another

mutually exclusive

two events cannot occur at the same time and have no outcomes in common -P (A ∪ B) = P (A) + P (B) -A and B are mutually exclusive only if P (A ∩ B) = 0