Exam 6: Counting

In questions a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible?

How many permutations of the seven letters A, B, C, D, E, F, G neither begin nor end with A?

In questions nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. -In how many ways can this be done if Ann and Ben are in the picture, but not standing next to each other?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, one of which is 3.

You have 20 pennies, 30 nickels, and 40 dimes. Assume that the pennies are identical, the nickels are identical, and the dimes are identical. In how many ways can you put all the coins in a row?

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words are there?

How many permutations of the seven letters A, B, C, D, E, F, G do not have vowels on the ends?

Find the largest coefficient in the expansion of $( x + 1 ) ^ { 6 }$

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with AAB in some order?

Use the binomial theorem to expand $( x + y ) ^ { 5 }$

Find the number of permutations of the letters in the word TATTERED.

$\text { Suppose } | A | = 10 \text { and } | B | = 4 \text {. Find the number of 1-1 functions } f : A \rightarrow B \text {. }$ A.

Find the coefficient of $x ^ { 9 }$ in the expansion of $\left( 2 + x ^ { 3 } \right) ^ { 10 }$

A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.

In questions a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible if the committee must consist of all women or all men?

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly five elements, including 3 or 4 but not both.

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (for example, two kings or two 5s). (b) three of a kind (for example, three 7s).

In questions nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. -In how many ways can this be done if neither Ed nor Fran are in the picture?

Find the number of permutations of the letters in your last name.

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with a vowel and end with a vowel?