Question

How do I find the coefficient using Pascal’s triangle?

Answer 1

In this question, we have to find the coefficient using Pascal’s triangle. As we know, Pascal's triangle is a triangular array of the binomial coefficients that arise in the binomial theorem. Also, binomial coefficients are those numbers that consist of “n choose r” . Thus, we will form a Pascal triangle, and thus get the coefficient of the binomial theorem, which is the required result for the problem.

Complete step-by-step solution:
According to the question, we have to find the coefficient using Pascal's triangle.
As we know, a binomial coefficient is expressed in the form of $\left( \dfrac{n}{r} \right)\text{ }or\text{ }{}^{n}{{C}_{r}}\text{ }or\text{ C(n,r) or C}_{r}^{n}$ .
In the Pascal triangle, the first row depicts 0, the second row depicts 1, and so on. The first column depicts 0, the second column depicts 1 and so on till the nth column.
Now, the binomial coefficients are used in the binomial theorem ${{(a+b)}^{n}}=\sum{\left( \begin{aligned} & n \\\ & k \\\ \end{aligned} \right)}{{a}^{n-k}}{{b}^{k}}$ .
Thus, $\left( \begin{aligned} & n \\\ & r \\\ \end{aligned} \right)$ depict the nth row and rth element in the Pascal triangle.
Thus, the Pascal triangle for the elements of the 5th row is given below
$\begin{aligned} & \text{0th row 1} \\\ & \text{1st row 1 1} \\\ & \text{2nd row 1 2 1} \\\ & \text{3rd row 1 3 3 1} \\\ & \text{4th row 1 4 6 4 1} \\\ & \text{5th row 1 5 10 10 5 1} \\\ & \text{ } \\\ & \text{ 0th 1st 2nd 3rd 4th 5th} \\\ \end{aligned}$
Thus, we will take an example to have more clarification, that is
Example: What is the coefficient of $\left( \begin{aligned} & 5 \\\ & 4 \\\ \end{aligned} \right)$ .
Thus, we know that n depicts the row and r depicts the element of the Pascal triangle, therefore $n=5$ and $r=4$ . We see from the Pascal triangle that in the 5th row, we have the element as 1, 5, 10, 10, 5, and 1. Therefore, the element that lies in the 4th column is the coefficient of $\left( \begin{aligned} & 5 \\\ & 4 \\\ \end{aligned} \right)$ , that is
$\left( \begin{aligned} & 5 \\\ & 4 \\\ \end{aligned} \right)=5$.

Note: While solving this problem, do mention all the formulas correctly to avoid the mathematical errors. Always remember how to draw the Pascal triangle because that is the base to get the required solution to the problem.