Question

In a men tournament each of the seven players will play every other player exactly once. How many matches will be played during the tournament?
A. 12
B. 21
C. 14
D. 13
E. 18

Answer 1

Hint: If we consider the one player then we have six players remaining with whom a match can happen. For the second player there are 5 players other than the first player with whom a match can happen. Similarly, we will find all the possible way to have a match between the players and then to get the total number of matches played.

Complete step-by-step answer:
We have been given that in a men tournament each of seven players will play every other player exactly once.
Let the seven players to be \[{{A}_{1}},{{A}_{2}},{{A}_{3}},{{A}_{4}},{{A}_{5}},{{A}_{6}}\text{ and }{{\text{A}}_{\text{7}}}\text{.}\]
Case I:
If we consider A, then we have a total 6 remaining players.
So, number of matches = 6
Case II:
If we consider \[{{A}_{2}}\] then we have total 5 remaining players other than \[{{A}_{1}}\]
So, number of matches = 5
Case III:
If we consider \[{{A}_{3}}\] then we have total 4 remaining players other than \[{{A}_{1}}\text{ and }{{\text{A}}_{\text{2}}}\]
So, number of matches = 4
Case IV:
If we consider \[{{A}_{4}}\] then we have total 3 remaining players other than \[{{A}_{1}},{{A}_{2}}\text{ and }{{\text{A}}_{\text{3}}}\]
So, number of matches = 3
Case V:
If we consider \[{{A}_{5}}\] then we have a total 2 remaining players.
So, number of matches = 2
Case VI:
If we consider \[{{A}_{6}}\] then we have only 1 i.e. \[{{A}_{7}}\]
So, number of matches = 1
Total number of match \[\Rightarrow 6+5+4+3+2+1\text{ =21}\]
So, the correct answer is “Option B”.

Note: In this type of question it is very important to visualize the problem. So, we should suppose the variable first and then think of the possible way that can be happened by taking the different cases.Also, be careful while considering \[{{A}_{2}},{{A}_{3}}\] and so on as you will think that for \[{{A}_{2}}\] there are six remaining players but we already take the \[{{A}_{1}}\] so we have to not include it and similarly for all the other cases.