Exam 6: A: Counting

consider all bit strings of length 12. -How many have exactly four 1's and none of these 1's are adjacent to each other?

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 have E in one of the first two positions?

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with A or end with B?

Find the next largest permutation in lexicographic order after 52143.

consider all bit strings of length 12. -How many begin with 11 and end with 10?

A game consisting of flipping a coin ends when the player gets two heads in a row, two tails in a row, or flips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words end with the letter T?

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

How many 5-element DNA sequences use all four bases, A, T, C, and G?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

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

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

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

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483 are strings in A. -How many strings in A have exactly one 5?

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly five elements, two of which are 3 and 4.

Use the binomial theorem to prove the following: $\left( \begin{array} { c } 100 \\0\end{array} \right) + \left( \begin{array} { c } 100 \\2\end{array} \right) + \left( \begin{array} { c } 100 \\4\end{array} \right) + \left( \begin{array} { c } 100 \\6\end{array} \right) + \cdots + \left( \begin{array} { c } 100 \\98\end{array} \right) + \left( \begin{array} { c } 100 \\100\end{array} \right) = \left( \begin{array} { c } 100 \\1\end{array} \right) + \left( \begin{array} { c } 100 \\3\end{array} \right) + \left( \begin{array} { c } 100 \\5\end{array} \right) + \cdots + \left( \begin{array} { c } 100 \\97\end{array} \right) + \left( \begin{array} { c } 100 \\99\end{array} \right)$

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

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 Ben is to be in the picture?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6.