Question

How do you evaluate ${}^6{C_4}?$

Answer 1

Hint: First of all, we have to address the formula and the values, then substitute the value of ‘n’ and ‘r’ in the formula and then find factorial for substituting values and then simply the expression.

Complete step by step solution:
We have been given that we have to find all the possible combinations for ${}^6{C_4}$. So first we will address the formula, input parameters, and values in the formula.
Then substitute the values of ‘n’ and ‘r’ in the formula and then we have to find the factorial for substituting values, then substitute the corresponding value in the formula.
According to the question:
Given that,
n= 6 and r= 4
so, the formula is,
$ \Rightarrow $${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Now, we have to substitute ‘n’ and ‘r’ values in the formula,
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( {6 - 4} \right)!}}$
Solving the above expression, we get –
$ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( 2 \right)!}}$
Now, we have to find the factorial for $6!$, $4!$, and $2!$,
$ \Rightarrow $$6! = $ $6 \times 5 \times 4 \times 3 \times 2 \times 1$
$ \Rightarrow $$4! = $ $4 \times 3 \times 2 \times 1$
$ \Rightarrow $$2! = $ $2 \times 1$
So, substitute the corresponding value in the above expression,
$ \Rightarrow $$^6{C_4} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{\left( {4 \times 3 \times 2 \times 1} \right)\left( {2 \times 1} \right)}}$
Now, we have to simply the above expression, we get –
$ \Rightarrow {}^6{C_4} = \dfrac{{6 \times 5}}{{2 \times 1}}$
Again, solving the above expression, we get –
$ \Rightarrow {}^6{C_4} = \dfrac{{30}}{2}$
Now, we can clearly see that 30 and 2 have the same common, so they can easily cancel out.
$ \Rightarrow {}^6{C_4} = 15$

So, 15 is the total possible combination for 6 CHOOSE 4
Hence, our required answer is 15.

Note:
In the case of combination, there is no order required but in permutation, the order is more important.
The product of all positive integers less than or equal to a given positive integer and denoted by the integer and an exclamation point. Thus factorial three is written $3!$ , meaning $3 \times 2 \times 1$ . The factorial of zero is defined as equal to one, $0! = 1$.