What is the coefficient of x9 in (x+2)¹5?

Solution :- x+215Apply binomial theorem: a+bn=∑i=0nnian-ibia=x, b=2=∑i=01515ix15-i· 2iExpand…

Answered: What is the coefficient of x9 in (x +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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What is the coefficient of x9 in (x+2)¹5?

Expert Solution

Step 1

Solution :-

${(x + 2)}^{15} Apply binomial theorem : {(a + b)}^{n} = \sum_{i = 0}^{n} (\binom{n}{i}) a^{(n - i)} b^{i} a = x, b = 2 = \sum_{i = 0}^{15} (\binom{15}{i}) x^{(15 - i)} \cdot 2^{i} Expand summation Note : - (\binom{n}{i}) = \frac{n!}{i! (n - i)!} = \frac{15!}{0! (15 - 0)!} x^{15} \cdot 2^{0} + \frac{15!}{1! (15 - 1)!} x^{14} \cdot 2^{1} + \frac{15!}{2! (15 - 2)!} x^{13} \cdot 2^{2} + \frac{15!}{3! (15 - 3)!} x^{12} \cdot 2^{3} + \frac{15!}{4! (15 - 4)!} x^{11} \cdot 2^{4} + \frac{15!}{5! (15 - 5)!} x^{10} \cdot 2^{5} + \frac{15!}{6! (15 - 6)!} x^{9} \cdot 2^{6} + \frac{15!}{7! (15 - 7)!} x^{8} \cdot 2^{7} + \frac{15!}{8! (15 - 8)!} x^{7} \cdot 2^{8} + \frac{15!}{9! (15 - 9)!} x^{6} \cdot 2^{9} + \frac{15!}{10! (15 - 10)!} x^{5} \cdot 2^{10} + \frac{15!}{11! (15 - 11)!} x^{4} \cdot 2^{11} + \frac{15!}{12! (15 - 12)!} x^{3} \cdot 2^{12} + \frac{15!}{13! (15 - 13)!} x^{2} \cdot 2^{13} + \frac{15!}{14! (15 - 14)!} x^{1} \cdot 2^{14} + \frac{15!}{15! (15 - 15)!} x^{0} \cdot 2^{15}$