Question

In a cricket championship there are 36 matches. The number of teams, if each plays 1 match with other, is:
A.9
B.10
C.8
D.12

Answer 1

Hint: In order to solve the problem first we will consider the number of teams in the championship as some unknown variables. Further we will use the method of combination to find the number of matches by finding the number of ways of selection of two teams out of the given number. Further we will solve the equation to find the value of the unknown variable.

Complete step by step answer:
Let us consider there are x number of teams in the championship.
Also we know that the number of matches in the championship is 36. And we have to find a number of teams.
Number of matches in the championship will be the same as the number of selections of two teams out of x teams.
As we know that number of ways selection of n different items from m number of different items is given as:
${}^m{C_n}$
Using the above formula number of ways of selection of two teams out of x teams is:
${}^x{C_2}$
The number of matches is 36.
So, let us equate the term and number of matches. So, we have:
$ \Rightarrow {}^x{C_2} = 36$
Let us simplify the above equation to find the value of x by using the formula of combination.
$
   \Rightarrow {}^x{C_2} = \dfrac{{x!}}{{\left( {x - 2} \right)! \times 2!}} = 36 \\
   \Rightarrow \dfrac{{x!}}{{\left( {x - 2} \right)! \times 2!}} = 36 \\
   \Rightarrow \dfrac{{x \times \left( {x - 1} \right)}}{2} = 36 \\
   \Rightarrow x\left( {x - 1} \right) = 72 = 9 \times 8 \\
   \Rightarrow x = 9 \\
 $
Hence, there are 9 teams in the championship.
So, option A is the correct answer.

Note: In order to solve such problems, students must remember the concept of permutation and combination along with formulas of factorial identity. Students must remember that we use combinations in order to find out the number of selections out of a given number of items and we use permutation in order to find the total number of arrangements out of the given number of items.