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Chapter 1: Problem 10

How many ways are there to pick a five-person basketball team from 12 possible players? How many selections include the weakest and the strongest players?

Short Answer

Expert verified
There are 792 ways to pick a 5-person basketball team out of 12 players. Out of these 792 teams, 120 of them will include both the strongest and weakest players.

Step by step solution

01

Calculation of Total Combinations

The first task is to determine how many ways are there to select 5 players out of 12. This is a problem of combination as the order of selection doesn't matter. The formula for calculating combinations is \(C(n, r) = \frac{n!}{r!(n - r)!}\) where \(n\) is the total number of items, \(r\) is the number of items to choose, and ! denotes the factorial function. Substituting the given values \(n = 12\) and \(r = 5\), we get \(C(12, 5) = \frac{12!}{5!(12 - 5)!}\)
02

Calculate Combinations including Strongest and Weakest Players

The second task is to find out the number of ways to select 5 players including both the weakest and the strongest players. As these two are already selected, it would mean to pick 3 out of the remaining 10 players. We can use the same formula to find you can have, \(C(10, 3) = \frac{10!}{3!(10 - 3)!}\)

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Most popular questions from this chapter

a) How many nonnegative integer solutions are there to the pair of equations \(x_{1}+x_{2}+x_{3}+\cdots+x_{7}=37, x_{1}+x_{2}+x_{3}=6\) ? b) How many solutions in part (a) have \(x_{1}, x_{2}, x_{3}>0\) ?

A committee of 12 is to be selected from 10 men and 10 women. In how many ways can the selection be carried out if (a) there are no restrictions? (b) there must be six men and six women? (c) there must be an even number of women? (d) there must be more women than men? (e) there must be at least eight men?

Write a computer program (or develop an algorithm) to determine whether there is a three-digit integer \(a b c(=100 a+10 b+c)\) where \(a b c=a !+b !+c !\)

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