Question

1. using pascals triangle, what is the coefficient of the $x^{2}y^{2}$ term in $(x + y)^{4}$?

a. 8
b. 3
c. 4
d. 6

Explanation:

Step1: Write out Pascal's Triangle

The first few rows of Pascal's Triangle are:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
The \(n\) -th row of Pascal's Triangle gives the coefficients of \((a + b)^n\). Here \(n = 4\), so we look at the 5th row (counting the first row as row 0).

Step2: Determine the position of the coefficient

For \((x + y)^n=\sum_{k = 0}^{n}C_{n}^kx^{n - k}y^{k}\), in the case of \((x + y)^4\), when \(n=4\) and we want the coefficient of \(x^{2}y^{2}\), we use the binomial - coefficient formula \(C_{n}^k=\frac{n!}{k!(n - k)!}\), or we can use Pascal's Triangle. In the expansion of \((x + y)^4=a_0x^{4}+a_1x^{3}y + a_2x^{2}y^{2}+a_3xy^{3}+a_4y^{4}\), the coefficients come from the 5th row of Pascal's Triangle: 1, 4, 6, 4, 1. The coefficient of \(x^{2}y^{2}\) is the third number in the 5th row (counting from 0), which is 6.

Answer:

D. 6