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Chapter 5: Problem 380

In chess championship 153 games have been played. If a player with every other player plays only once, then the number of players are (a) 17 (b) 51 (c) 18 (d) 35

Short Answer

Expert verified
The number of players in the chess championship is \(18\). Therefore, the correct answer is (c) 18.

Step by step solution

01

Understand the concept of combinations

In combinatorics, a combination is a selection of items from a larger set, such that the order of the items does not matter. In this case, we are looking for the number of unique pairs of players that can be formed from the total number of players. The formula for combinations is: \[C(n, k) = \frac{n!}{k!(n-k)!}\] Where \(n\) is the total number of items (in our case, the number of players), \(k\) is the number of items in each combination (in this case, 2, since we are looking for unique pairs), and \(C(n, k)\) is the number of combinations.
02

Find the number of players corresponding to the number of games played

Given that 153 games have been played, our goal is to find a value of \(n (the number of players)\) such that \(C(n, 2) = 153\), using the combinations formula from step 1. Rewriting the equation with the formula, we get: \[\frac{n!}{2!(n-2)!} = 153\] We need to find the value of \(n\) that satisfies this equation.
03

Solve for n

We will now solve the equation for \(n\): \[\frac{n!}{2!(n-2)!} = 153\] We know that \((n-1)! = \frac{n!}{n}\), so we can rewrite the equation as: \[\frac{n(n-1)}{2} = 153\] Now, multiply both sides by 2: \[n(n-1) = 306\] Since 17 * 18 = 306, \(n = 18\).
04

Select the correct answer

The correct answer is the number of players who participated in the chess championship: 18 players. Thus, the answer is (c) 18.

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