Question

"Find the coefficient of y⁵ in the expansion of (3+2y)⁸ .

A. 3360
B. 6720
C. 1680
D. 420"

Answer 1

The coefficient of
`\(y^5\)` in the expansion of
`\((3+2y)^8\)` is 6720 (Option B).

Step-by-step explanation:

To find the coefficient of
`\(y^5\)` in the expansion of
`\((3+2y)^8\)`, we use the binomial theorem formula. The general term in the expansion is given by
`\(T_k = \binom{8}{k} \cdot (3)^(8-k) \cdot (2y)^k\)`. To find the term with
`\(y^5\)`, we set (k = 5), and the coefficient is
`\(\binom{8}{5} \cdot (3)^3 \cdot (2)^5\)`.

`\(\binom{8}{5} = (8!)/(5!(8-5)!) = (8 \cdot 7 \cdot 6)/(3 \cdot 2 \cdot 1) = 56\)`.

So, the coefficient of
`\(y^5\)` is
`\(56 \cdot 3^3 \cdot 2^5 = 6720\)`.

The binomial theorem is a powerful tool for expanding expressions of the form
`\((a+b)^n\)`. It provides a systematic way to find the coefficients of the terms in the expansion. In this case, applying the binomial theorem with
`\(a = 3\)` and
`\(b = 2y\)`, we identified the term with
`\(y^5\)` and calculated its coefficient. The correct answer is 6720, corresponding to Option B.