Compute the following:

• [math]{6 \choose 3}[/math]
• [math]b(5,.2,4)[/math]
• [math]{7 \choose 2}[/math]
• [math]{{26} \choose {26}}[/math]
• [math]b(4,.2,3)[/math]
• [math]{6 \choose 2}[/math]
• [math]{{10} \choose 9}[/math]
• [math]b(8, .3, 5)[/math]

In how many ways can we choose five people from a group of ten to form a committee?

How many seven-element subsets are there in a set of nine elements?

Using the relation Equation 3.1

write a program to compute Pascal's triangle, putting the results in a matrix. Have your program print the triangle for [math]n = 10[/math].

Use the program BinomialProbabilities to find the probability that, in 100 tosses of a fair coin, the number of heads that turns up lies between 35 and 65,between 40 and 60, and between 45 and 55.

Charles claims that he can distinguish between beer and ale 75 percent of the time. Ruth bets that he cannot and, in fact, just guesses. To settle this, a bet is made: Charles is to be given ten small glasses, each having been filled with beer or ale, chosen by tossing a fair coin. He wins the bet if he gets seven or more correct. Find the probability that Charles wins if he has the ability that he claims. Find the probability that Ruth wins if Charles is guessing.

Show that

for [math]j \ge 1[/math]. Use this fact to determine the value or values of [math]j[/math] which give [math]b(n,p,j)[/math] its greatest value. Hint: Consider the successive ratios as [math]j[/math] increases.

A die is rolled 30 times. What is the probability that a 6 turns up exactly 5 times? What is the most probable number of times that a 6 will turn up?

Find integers [math]n[/math] and [math]r[/math] such that the following equation is true:

In a ten-question true-false exam, find the probability that a student gets a grade of 70 percent or better by guessing. Answer the same question if the test has 30 questions, and if the test has 50 questions.