Question

In a tennis tournament, each of the six players play with every other player exactly once. How many matches are played during the tournament?
(a) 12
(b) 30
(c) 36
(d) 15

Answer 1

Hint: In order to find the solution of this question, we should know about the concept of the combination, that is whenever we have to choose a few out of some, then we have to apply the formula, that is, $${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$$ where r is the number of things we have to choose out of n and n is the total things.

Complete step-by-step answer:
In this question, we have been asked to find the number of matches that will be played by 6 players such that each player plays with every other for only once. To solve this question, we should know about the concept of the combination, that is, if we have to choose r out of n, then we can say that the possible number of combinations are, $${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}.$$
Now, we know that at a time, only 2 players will play out of 6. So, we can say that the number of ways will be equal to the number of ways of choosing 2 out of 6. So, by the concept of combination, we can say, the total number of matches are $${}^6{C}_2.$$
And by the formula of the combination, that is $${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}},$$ we can write the total number of matches as
$${}^6{C}_2={\displaystyle \frac{6!}{2!(6-2)!}}$$
Now, we will simplify it further. So, we get, the total number of matches as,
$$\Rightarrow {\displaystyle \frac{6!}{2!4!}}$$
$$\Rightarrow {\displaystyle \frac{6\times 5}{2}}$$
= 15
Hence, we can say 6 players will play 15 matches in total such that each player will play with every other player.
Therefore, we can say that option (d) is the right answer.

Note: While solving this question, one can think of making pairs by taking 2 players at a time but that will waste our time. We may miss a few cases which will give us a wrong answer and we may lose our marks. So, it is better to use the formula of combinations to solve the question.