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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) If \(n\) and \(r\) are positive integers with \(n \geq r\), how many solutions are there to $$ x_{1}+x_{2}+\cdots+x_{r}=n $$ where each \(x_{i}\) is a positive integer, for \(1 \leq i \leq r ?\) b) In how many ways can a positive integer \(n\) be written as a sum of \(r\) positive integer summands \((1 \leq r \leq n)\) if the order of the summands is relevant?

a) In how many ways can seven people be arranged about a circular table? b) If two of the people insist on sitting next to each other, how many arrangements are possible?

The production of a machine part consists of four stages. There are six assembly lines available for the first stage, four assembly lines for the second stage, five for the third stage, and five for the last. Determine the number of different ways in which a machine part can be totally assembled in this production process.

a) In how many ways can we select five coins from a collection of 10 consisting of one penny, one nickel, one dime, one quarter, one half-dollar, and five (identical) Susan B. Anthony dollars? b) In how many ways can we select \(n\) objects from a collection of size \(2 n\) that consists of \(n\) distinct and \(n\) identical objects?

Four numbers are selected from the following list of numbers: \(-5,-4,-3,-2,-1,1,2,3,4\). (a) In how many ways can the selections be made so that the product of the four numbers is positive and (i) the numbers are distinct? (ii) each number may be selected as many as four times? (iii) each number may be selected at most three times? (b) Answer part (a) with the product of the four numbers negative.
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