In the expansion of (1+x)^n(1+y)^n(1+z)^n, the sum of the co... | Mathematics - Properties of Binomial Coefficients | SolveeIt

Question

In the expansion of $(1+x)^n(1+y)^n(1+z)^n$ , the sum of the co-efficients of the terms of degree 'r' is

${}^nC_r$

${}^nC_r^3$

${}^{3n}C_r$

3. ${}^{2n}C_r$

Answer

${}^{3n}C_r$

Explanation

Solution

The sum of coefficients of terms of degree 'r' in a polynomial $P(x_1, x_2, \dots, x_m)$ is equal to the coefficient of $t^r$ in the polynomial $P(t, t, \dots, t)$ .

Given the expression $P(x,y,z) = (1+x)^n(1+y)^n(1+z)^n$ . To find the sum of coefficients of terms of degree 'r', we substitute $x=t, y=t, z=t$ : $P(t,t,t) = (1+t)^n (1+t)^n (1+t)^n = (1+t)^{3n}$ .

Now, we expand $(1+t)^{3n}$ using the binomial theorem: $(1+t)^{3n} = \sum_{r=0}^{3n} {}^{3n}C_r t^r$ .

The coefficient of $t^r$ in this expansion is ${}^{3n}C_r$ . Therefore, the sum of the coefficients of the terms of degree 'r' in the original expansion is ${}^{3n}C_r$ .

Question: In the expansion of $(1+x)^n(1+y)^n(1+z)^n$, the sum of the co-efficients of the terms of degree 'r'...

Solution