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The largest term in the binomial expansion of $$(1+\dfrac{1}{2})^{31}$$  is of the form $$\dfrac{a}{b}$$, where  $$a$$ and  $$b$$ are relatively prime positive integers. What is the value of $$b$$?

Mar 29, 2024

#1
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The largest term in the binomial expansion of (a+b)n occurs when k = n/2 (for even n) or k = (n - 1)/2 (for odd n) in the binomial coefficient \binom{n}{k} = \frac{n!}{k!(n-k)!}.

In this case, n = 31 (odd). Therefore, the largest term occurs when k = (31 - 1)/2 = 15.

The term we want is \binom{31}{15} = \frac{31!}{15!16!}. We can re-write this as:

\binom{31}{15} = \frac{31 \times 30 \times 29 \times \cdots \times 16}{16 \times 15 \times 14 \times \cdots \times 1}

Notice that most of the terms in the numerator and denominator cancel out, leaving us with:

\binom{31}{15} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1} = 15450

The denominator, b, is thus 3 \times 2 \times 1 = \boxed{6}.

Mar 29, 2024
#2
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Thank you so much!

Mar 29, 2024