Exam 6: Counting

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

In questions consider all bit strings of length 12. -How many begin with 110?

(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?

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies.

In questions let A be the set of all strings of decimal digits of length five. For example 00312 and 19483 are strings in -Find |A|.

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

Findodd. the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is

In how many ways can you get a bunch of four books to give to a friend?

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

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 or end with A or B?

The figure at the right shows a 4-block by 5-block grid of streets. to point B, where at each stage you can only go right or up. Find the number of ways in which you can go from point A (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.

Find the next largest permutation in lexicographic order after 3254781.

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

Find the largest coefficient in the expansion of $( x + 3 ) ^ { 5 }$ .

Find the next largest permutation in lexicographic order after 1324.

In questions 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 both 5 and 6.

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

In questions suppose you have a class with 30 students - 10 freshmen, 12 sophomores, and 8 juniors. -In how many ways can you put all students in a line so that the freshmen are first, the sophomores are in the middle, and the juniors are at the end?

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