Question

How do you calculate combinations?

Answer 1

Hint: We have to calculate combinations as per the question. Combinations are used to count the number of arrangements possible in a collection of items. But the order is not important in combinations. The formula for combinations is \[\dfrac{n!}{r!(n-r)!}\], where ‘n’ refers to the total number of possibilities, ‘r’ refers to the number of selections made.

Complete step-by-step solution:
According to the given question, we have to write about how to calculate combinations.
Combinations is a mathematical concept which is used to count the number of arrangements possible which is possible in a collection of items. The important thing to keep in mind is that, in combinations the order in which arrangement is done is not of any concern, that means order is not significant in combinations.
The formula to find the combinations is \[^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\], where ‘n’ refers to the total number of possibilities, ‘r’ refers to the number of selection made.
For example – Suppose we have a group of 5 people and we have to select 3 from the group.
So the number of ways the selection is possible will be found by using combinations as order is not of significance here.
The formula we have, \[^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}\].
Here, \[n=5\] and \[r=3\]
Substituting the values in the formula for combination, we get,
\[^{5}{{C}_{3}}=\dfrac{5!}{3!(5-3)!}\]
Solving further we get,
\[\Rightarrow \dfrac{5!}{3!2!}\]
\[\Rightarrow \dfrac{5\times 4\times 3!}{3!2!}\]
Cancelling the similar factorial, we get,
\[\Rightarrow \dfrac{5\times 4}{2!}\]
\[\Rightarrow \dfrac{5\times 4}{2}\]
\[\Rightarrow 5\times 2=10\]
So, there are 10 ways in which the selection can be done.

Note: Along with Combinations, we also have permutations. While order is not important in combinations but in permutation order is of significance. So, while answering the questions based on the number of possibilities, always check if order is important or not and then accordingly use either permutation or combination.
Permutation is represented as follows,
\[^{n}{{P}_{r}}=\dfrac{n!}{(n-r)!}\] where ‘n’ refers to the total number of possibilities, ‘r’ refers to the number of selection made.