Exam 6: A: Counting

let A be the set of all bit strings of length 10. -How many bit strings of length 10 have exactly six 0's?

Show that if five points are picked on or in the interior of a square of side length 2 , then there are at least two of these points no farther than $\sqrt { 2 }$ apart.

Find the number of terms in the expansion of $( 5 a + 8 b ) ^ { 15 }$

Use the binomial theorem to prove the following: $\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 } .$

Use the binomial theorem to expand $( a + 2 ) ^ { 6 }$

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

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 three 5s?

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 5_S). (b) three of a kind (for example, three 7_S).

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?

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

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

Here is an incorrect solution to a problem. Find the error, explain why it is not correct, and give the correct answer. "Problem: Find the number of ways to get two pairs of two different ranks (such as 2 jacks and 2 fives) in a 4-card hand from an ordinary deck of 52 cards." "Solution: There are 13 ways to get a rank (such as "kings") for the first pair and ( $\left. \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get a pair of that rank. Similarly, there are 12 ways to get a rank (such as "sevens") for the second pair and $\left( \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get a pair of that rank. Therefore there are $13 \cdot \left( \begin{array} { l } 4 \\ 2 \end{array} \right) \cdot 12 \cdot \left( \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get 2 pairs."

How many permutations of the seven letters A, B, C, D, E, F, G do not have the vowels next to each other?

let A be the set of all bit strings of length 10. -How many bit strings of length 10 are there?

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.