Combinatorics Actual Examples

Find the number of ways the word 'MISCELLANEOUS' can be arranged such that the vowels always come together.

3268800

Given set A = {8, 4, 3, 5, 0, 9}, how many 3-digit numbers can be formed and which are divisible by 5 and none of the digits can be repeated?

36

How many ways can the letters of the word 'SOCIAL' be arranged such that the consonants must occupy only the odd positions?

36

There are 20 tricycles you can ride from Dorm to CAS or vice versa. In how many ways can a person go from Dorm to CAS and return by a different tricycle?

380

x1+x2+x3=11 x1<=1 x2<=2 x3<=3

*|**|*** 11-6, distribute 5 so (2+5, 5)

P(n, 0) =

1

Give the set of digits 2, 5, 4, 0, 7, 8, how many numbers between 100 and 1000 can be formed if none of the digits can be repeated?

100

On a fast dating camp organized by me, I have invited 10 single men and single 10 women to participate in the event. In how many ways can I make a pair of a man and a woman?

100

Out of digits 5, 6, 7, 8, how many 2 digit numbers can be generated without repeating any digit?

12

the english alphabet contains 21 consonants and 5 vowels. how many strings of 6 lowercase letters of the alphabet contain exactly one vowel exactly two vowels at least one vowel at least two vowels

21^5 * 6 * 5 5 vowels can put into 6 places and 21^5 to place the rest of the consonants

In a seminar hosted by ASEAN, every person shakes hand with every other person. If there was a total of 28 handshakes in the party, how many persons were present during the seminar?

8

What will be the number of possible outcomes when a coin is tossed 3 times?

8

In how many ways can we select three students from a group of five students to stand in line for a picture? In how many ways can we arrange all five of these students in a line for a picture?

: First, note that the order in which we select the students matters. There are five ways to select the first student to stand at the start of the line. Once this student has been selected, there are four ways to select the second student in the line. After the first and second students have been selected, there are three ways to select the third student in the line. By the product rule, there are 5 · 4 · 3 = 60 ways to select three students from a group of five students to stand in line for a picture. To arrange all five students in a line for a picture, we select the first student in five ways, the second in four ways, the third in three ways, the fourth in two ways, and the fifth in one way. Consequently, there are 5 · 4 · 3 · 2 · 1 = 120 ways to arrange all five students in a line for a picture

how many ways are there for 10 women and 6 men to stand in line so that no two men stand next to each other

P(8,8)*P(9,5) consider the spaces in between

100 tickets are sold to 100 different people. 4 prizes are awarded. how many ways are there to award the prizes if the person holding ticket 47 wins one of the prizes

P(99, 3) * 4 since person 47 wins either of the 4 prizes so times 4 what if daog ya ang first te P(99,3) what if ang second P(99,3) man gyapon or ang third P(99,3) 4th P(99,3)

How many bit strings of length 10 contain a.) exactly four 1s? b.) at most four 1s? c.) at least four 1s? d.) an equal number of 0s and 1s

a.) C(10, 4) explanation: to fill the 1s C(10, 4). to fill the remaining slots with zero C(6, 6). same other way around b.) C(10, 0) + C(10, 1) + C(10, 2) + C(10, 3) c.) 2^10 - b d.) C(10, 5)

p(n, n)

n!

general formula

p(n, r) = C(n, r) * P(r, r)