Exam 6: A: Counting

Find the next largest permutation in lexicographic order after 6714235.

The figure at the right shows a 4-block by 5-block grid of streets. Find the number of ways in which you can go from point A to point B , where at each stage you can only go right or up. (You are not allowed to go left or down.) For example, one allowable route from A to B is: Right, Right, Up, Right, Up, Up, Right, Right, Up.

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?

Find the number of positive integers not exceeding 1000 that are not divisible by 4, 6, or 9.

Find the next largest permutation in lexicographic order after 3254781.

Make up a word problem in good English whose answer is $\left( \begin{array} { c } 15 \\4\end{array} \right) \cdot \left( \begin{array} { l } 7 \\3\end{array} \right)$

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

A class consists of 20 sophomores and 15 freshmen. The club needs to choose four different members to be president, vice president, secretary, and treasurer. (a) In how many ways is this possible? (b) In how many ways is this possible if sophomores will be chosen as president and treasurer and freshmen as vice president and secretary?

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

You have 50 of each of the following kinds of jellybeans: red, orange, green, yellow. The jellybeans of each color are identical. (a) In how many ways can you put all the jellybeans in a row? (b) How many handfuls of 12 are possible?

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

(a) In how many ways are there to arrange the letters of the word NONSENSE ? (b) How many of these ways start or end with the letter O?

Use the binomial theorem to expand $( 2 c - 3 d ) ^ { 4 }$

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

If the permutations of 1, 2, 3, 4, 5, 6 are written in lexicographic order, with 123456 in position #1, 123465 in position #2, etc., find the permutation in position #483.

Use the binomial theorem to expand $\left( x - \frac { 3 } { x } \right) ^ { 5 }$

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?

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

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all 30 books are different. -In how many ways can you get a bunch of four books to give to a friend?

assume that you have 50 pennies and three jars, labeled A, B, and C. -In how many ways can you put the pennies in the jars, assuming that the pennies are identical and each jar must have an even number of pennies put into it?