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The coefficients corresponding to k = 0, 1, 2, ..., 5 in the expansion of (x + y)^5 are _____.

A1, 5, 10, 10, 5, 1
B1, 5, 10, 5, 1
C0, 1, 5, 10, 5, 1, 0
D0, 5, 10, 10, 5, 0

Verified Answer :

MissPhiladelphia

The coefficients corresponding to k = 0, 1, 2, ..., 5 in the
expansion of (x + y)^5 are 1, 5 , 10 , 10, 5, 1. The coefficients can be solve
using manual expansion of the term of using combination:

nCr-1, where n is the power and r is the desired
therm,

Answer: A. 1, 5, 10, 10, 5, 1

Step-by-step explanation:

By the binomial expansion,

\((x+y)^n=\sum_{k=0}^{n} ^nC_k (x)^{n-k}(y)^k\)

Where,

\(^nC_k = \frac{n!}{k!(n-k)!}\)

Here, n = 5 and k = 0,1,2,......, 5.

Thus,

\((x+y)^5=\sum_{k=0}^{5} ^5C_k (x)^{n-k}(y)^k\)

\(= ^5C_0 (x)^{5-0}y^0+^5C_1 (x)^{5-1}y^1+^5C_2 (x)^{5-2}y^2+^5C_3 (x)^{5-3}y^3+^5C_4 (x)^{5-4}y^4+^5C_5 (x)^{5-5}y^5\)

\(= 1 x^ 5 + 5 x^4 y ^1 + 10 x^3y^2+ 10 x^2y^3 + 5 x^1y^4 + 1 y^5\)     ( Because, \(a^0 = 1\) )

Thus, the coefficients are 1, 5, 10, 5, 1.

⇒ Option A is correct.

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