If Cr stands for nCr, then the coefficient of λ nμ n in the expansion of [(1 + λ ) (1 + μ ) (λ + μ )]n is:

Question

If Cr stands for nCr, then the coefficient of λ nμ n in the expansion of [(1 + λ ) (1 + μ ) (λ + μ )]n is: (A)  displaystyle ∑ r = 0nCr2 (B)  displaystyle ∑ r = 0nCr + 22 (C)  displaystyle ∑ r = 0nCr + 32 (D)  displaystyle ∑ r = 0nCr3

Tardigrade Admin · Accepted Answer

(1 + λ )n(μ + 1)n(λ + μ )n= displaystyle ∑ r = 0nCr(λ )r displaystyle ∑ s = 0nCs(μ )n - s displaystyle ∑ t = 0nCt(λ )n - 1(μ )t since, C03 is the coefficient of λ 0μ n - 0λ n - 0μ 0 i.e., (λ )n(μ )n(r = s = t = 0) Now, C13 is the coefficient of λ 1μ n - 1λ n - 1μ 1 i.e., (λ )n(μ )n(r = s = t = 1) etc. and Ck3 is the coefficient of λ kμ n - kλ n - kμ k i.e., (λ )n(μ )n(r = s = t = k) Hence, the coefficient of λ nμ n in (1 + λ )n(μ + 1)n(λ + μ )n=C03+C13+C23+ ldots +Cn3= displaystyle ∑ r = 0nCr3

Q. If $C_{r}$ stands for $_{n} C_{r},$ then the coefficient of $λ^{n} μ^{n}$ in the expansion of $[(1 + λ) (1 + μ) (λ + μ)]^{n}$ is:

$r = 0 \sum n C_{r}^{2}$

$r = 0 \sum n C_{r + 2}^{2}$

$r = 0 \sum n C_{r + 3}^{2}$

$r = 0 \sum n C_{r}^{3}$

Q. If Cr​ stands for n​Cr​, then the coefficient of λnμn in the expansion of [(1+λ)(1+μ)(λ+μ)]n is:

r=0∑n​Cr2​

r=0∑n​Cr+22​

r=0∑n​Cr+32​

r=0∑n​Cr3​

Q. If $C_{r}$ stands for $_{n} C_{r},$ then the coefficient of $λ^{n} μ^{n}$ in the expansion of $[(1 + λ) (1 + μ) (λ + μ)]^{n}$ is:

$r = 0 \sum n C_{r}^{2}$

$r = 0 \sum n C_{r + 2}^{2}$

$r = 0 \sum n C_{r + 3}^{2}$

$r = 0 \sum n C_{r}^{3}$