The coefficient of x53 in the following expansions. is

Engineering

Mathematics

Introduction to Binomial Theorem

Multinomial Theorem

Application of Binomial Theorem and Summation Of Series

Question

The coefficient of x⁵³ in the following expansions. $\sum_{m = 0}^{100} {}^{100}C_{m} {(x - 3)}^{100 - m} . 2^{m}$ is

¹⁰⁰C₄₇

¹⁰⁰C₅₃

^{– 100}C₁₀₀

^{– 100}C₅₃

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$\sum_{m = 0}^{100} {}^{100}C_{m} {(x - 3)}^{100 - m} . 2^{m}$
Above expansion can be rewritten as

[(x – 3) + 2]¹⁰⁰ = (x – 1)¹⁰⁰ = (1 – x)¹⁰⁰
∴ ⁵³ will occur in T₅₄ i.e. 54^th term.

So, m = 53
⇒ T₅₄ = ¹⁰⁰C₅₃(– x)⁵³
∴ Required coefficient is ^–100C₅₃.