Question

Use the binomial Theorem to find the coefficient of x7 in the expansion of (2x + 1)12

Answer 1

Answer:

101,376

Step-by-step explanation:

The binomial theorem states that

$(a+b)^n=\sum\limits_{k=0}^nC^n_ka^{n-k}b^k$

Apply this theorem to the expression

$(2x+1)^{12}.$

We need only to know coefficient at $x^7,$ so

$n-k=7\\ \\12-k=7\\ \\k=12-7\\ \\k=5$

For k = 5, we have term in binomial expansion

$C^{12}_5\cdot(2x)^{12-5}\cdot 1^5\\ \\=\dfrac{12!}{5!(12-5)!}\cdot (2x)^7\cdot 1\\ \\=\dfrac{12!}{5!\cdot 7!}\cdot 2^7\cdot x^7\\ \\=\dfrac{7!\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 7!}\cdot 128x^7\\ \\=4\cdot 3\cdot 2\cdot 11\cdot 3\cdot 128x^7\\ \\=101,376x^7$