Question

How do you use the binomial theorem to calculate \[{}^6{C_4}\]?

Answer 1

Hint: In this question we have to find the value of the given combination, for doing this we will make use of the combination formula which is given by \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] where n is equal to the size of the set, r is equal to the size of the each combination, and ‘!’ is equal to the factorial operation, and by substituting the values in the formula we will get the required value of the combination.

Complete step-by-step answer:
Given expression is a combination \[{}^6{C_4}\],
The combination expression \[{}^n{C_r}\] is known as counting formula or combination formula. This formula can be used to count the number of possible combinations in a given situation.
The combination formula is given by \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\],
Now using the formula, here \[n = 6\] and \[r = 4\], so substituting the values in the formula we get,
\[ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( {6 - 4} \right)!}}\],
Now simplifying we get,
\[ \Rightarrow {}^6{C_4} = \dfrac{{6!}}{{4!\left( 2 \right)!}}\],
Now using factorial operation which is given by \[n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times \left( {n - 3} \right) \times ..............1\], we get,
\[ \Rightarrow {}^6{C_4} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{\left( {4 \times 3 \times 2 \times 1} \right)\left( {2 \times 1} \right)}}\],
Now simplifying by eliminating the like terms in both numerator and denominator we get,
\[ \Rightarrow {}^6{C_4} = \dfrac{{6 \times 5}}{{2 \times 1}}\],
Now again simplifying we get,
\[ \Rightarrow {}^6{C_4} = 3 \times 5\],
Now multiplying we get,
\[ \Rightarrow {}^6{C_4} = 15\],
So, the value of\[{}^6{C_4}\]is 15.

\[\therefore \]The value of \[{}^6{C_4}\] will be equal to 15.

Note:
The combination formula in maths shows the number of ways a given sample of “r” elements can be obtained from a larger set of “n” distinguishable number of objects.
Hence if the order doesn’t matter then we have a combination, and if the order does not matter then we have a permutation. Also we can say that a permutation is an ordered combination. And to use the combination formula we need to calculate the factorial of a number which is defined as the product of all the positive integers which is equal to and less than the number.