Exam 6: Counting

Find the coefficient of $x ^ { 5 } y ^ { 6 }$ in the expansion of $( 2 x - y ) ^ { 11 }$

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

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that have exactly one B.

Find the next largest permutation in lexicographic order after 6714235.

Find the next four largest 4-combinations of the set {1, 2, 3, 4, 5, 6, 7, 8} after {1, 2, 3, 5}.

assume that you have a bowl containing hard candies: 50 cherry, 50 strawberry, 40 orange, 70 lemon, and 40 pineapple. Assuming that the pieces of each flavor are identical, -How many handfuls of 15 are possible with at least three pieces of each flavor?

How many permutations of the seven letters A, B, C, D, E, F, G have the two vowels before the five consonants?

In questions let A be the set of all bit strings of length 10. -How many bit strings of length 10 have equal numbers of 0's and 1's?

A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

Find the next largest permutation in lexicographic order after 52143.

How many 8-element DNA sequences contain exactly four C's?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, but neither 3 nor 4.

assume that you have a bowl containing hard candies: 50 cherry, 50 strawberry, 40 orange, 70 lemon, and 40 pineapple. Assuming that the pieces of each flavor are identical, -How many handfuls of 15 are possible with at least two pieces of each flavor?

In questions 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 computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers.

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin with L or end with R.

Find the coefficient of $x ^ { 7 } y ^ { 5 }$ in the expansion of $( 3 x - y ) ^ { 12 }$

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

A factory makes automobile parts. Each part has a code consisting of a digit, a letter, and a digit, with the digits distinct, such as 5C7, 1O6, or 3Z0. Last week the factory made 5,000 parts. Find the minimum number of parts that must have the same serial number.

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 or B?