Question

How do you evaluate ${}^{11}{C_4}$?

Answer 1

Here, we will use the concept of combination. We will substitute the given values in the formula of combination and simplify the equation to get the required answer. The combination is defined as a method or way of arranging elements from a set of elements such that the order of arrangement does not matter.

Formula Used:
Combinations is given by the formula ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$

Complete step by step solution:
We are given a binomial coefficient ${}^{11}{C_4}$.
Now, we will use the combinations formula to find the value of the Binomial Coefficients.
Combinations is given by the formula ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
By substituting $n = 11$ and $r = 4$ in the combination formula, we get
${}^{11}{C_4} = \dfrac{{11!}}{{4!\left( {11 - 4} \right)!}}$
Now, by simplifying the expression, we get
$\Rightarrow {}^{11}{C_4} = \dfrac{{11!}}{{4!7!}}$
Now, by using the concept of Factorial, we get
$\Rightarrow {}^{11}{C_4} = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{4 \times 3 \times 2 \times 1 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}$
Now, by cancelling the terms, we get
$\Rightarrow {}^{11}{C_4} = \dfrac{{11 \times 10 \times 9 \times 8}}{{4 \times 3 \times 2 \times 1}}$
Now, by simplifying the terms, we get
$\Rightarrow {}^{11}{C_4} = 11 \times 10 \times 3$
$\Rightarrow {}^{11}{C_4} = 330$

Therefore, the value of ${}^{11}{C_4}$ is $330$.

Note:
We should know that the binomial coefficient uses the concept of combinations. Binomial coefficients are the integers which are coefficients in the Binomial Theorem. A permutation is defined as the arrangement of letters, numbers, or some elements in a set. It gives us the number of ways that the elements in a set are arranged. Both combination and permutation are used to arrange the elements but in permutations, the order is important while in combinations order is not important. Factorial is defined as the numbers multiplied in descending order till unity. Binomial coefficients are widely used in the binomial expansion. Pascal’s triangle is used to find the binomial coefficients.