/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Find the coefficient of \(t^{47}... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 53

Find the coefficient of \(t^{47}\) in the expansion of \((t+2)^{50}\).

Short Answer

Expert verified
The coefficient of \(t^{47}\) in the expansion of \((t+2)^{50}\) is \(\boxed{11200}\).

Step by step solution

01

Identify the corresponding value for r in the term \(t^{47}\)

In the expansion, the term we are looking for is of the form \(t^{50-r}\cdot2^r\). We need to find the value of \(r\) for which the term is \(t^{47}\). Since \(50-r=47\), we can conclude that \(r=3\). This means our term of interest is \(t^{47}\cdot2^3\).
02

Compute the binomial coefficient

The general term in the expansion of \((t+2)^{50}\) follows the binomial theorem, which states that it is of the form \(\binom{n}{r}t^{n-r}2^r\). In this case, \(n = 50\) and we found out that \(r = 3\). Therefore, we need to compute the binomial coefficient, \(\binom{50}{3}\). Using the formula \(\binom{n}{r}=\frac{n!}{r!(n-r)!}\), we have: \[ \binom{50}{3} = \frac{50!}{3!(50-3)!} = \frac{50!}{3!47!}. \]
03

Calculate the factorial values

We will now calculate the factorial values for \(50!\), \(3!\), and \(47!\). Recall that factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). \begin{align*} 50! &= 50 \cdot 49 \cdot 48 \cdot 47!\\ 3! &= 3 \cdot 2 \cdot 1 = 6 \\ 47! &= \text{(we leave this unchanged, as it will simplify in the division step)} \end{align*}
04

Simplify the binomial coefficient

Now, we can simplify the binomial coefficient by substituting the factorial values we calculated: \[ \binom{50}{3} = \frac{50!}{3!47!} = \frac{50 \cdot 49 \cdot 48 \cdot 47!}{6 \cdot 47!} = \frac{50 \cdot 49 \cdot 48}{6}. \] Cancel out any common factors between the numerator and the denominator: \[ \binom{50}{3} = \frac{50 \cdot 7 \cdot 8}{2} = 50 \cdot 7 \cdot 4 = 1400. \]
05

Multiply the binomial coefficient by the term \(t^{47}2^r\)

Finally, we need to multiply the binomial coefficient \(\binom{50}{3}=1400\) by the term \(t^{47}2^3\). Recall that \(r=3\): \[ \text{Coefficient of } t^{47} \text{ in the expansion} = 1400 \cdot t^{47} \cdot 2^3 = 1400 \cdot t^{47} \cdot 8. \] This simplifies to: \[ \text{Coefficient of } t^{47} \text{ in the expansion} = 11200 \cdot t^{47}. \] Thus, the coefficient of \(t^{47}\) in the expansion of \((t+2)^{50}\) is \(\boxed{11200}\).

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