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Given that $$\binom{40}{4} = \frac{40!}{4!b!}$$, (a) write down the value of b - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 3

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Given-that---$$\binom{40}{4}-=-\frac{40!}{4!b!}$$,---(a)-write-down-the-value-of-b-Edexcel-A-Level Maths Pure-Question 7-2011-Paper 3.png

Given that $$\binom{40}{4} = \frac{40!}{4!b!}$$, (a) write down the value of b. In the binomial expansion of $(1+x)^{40}$, the coefficients of $x^4$ and $x^s$... show full transcript

Worked Solution & Example Answer:Given that $$\binom{40}{4} = \frac{40!}{4!b!}$$, (a) write down the value of b - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 3

Step 1

write down the value of b.

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Answer

To find the value of bb, we start by using the formula for combinations:
(nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}.
For this problem:

  • Here, n=40n = 40 and r=4r = 4.
  • Thus, we have:
    (404)=40!4!(40−4)!=40!4!⋅36!\binom{40}{4} = \frac{40!}{4!(40-4)!} = \frac{40!}{4! \cdot 36!}.
  • Comparing this with the given equation:
    (404)=40!4!b!\binom{40}{4} = \frac{40!}{4! b!}, we can conclude that b=36b = 36.

Step 2

Find the value of \frac{q}{p}.

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Answer

For the binomial expansion of (1+x)40(1 + x)^{40}:

  • The coefficient of x4x^4 (which is pp) is given by:
    p=(404)=40!4!⋅36!=40×39×38×374×3×2×1=91,390.p = \binom{40}{4} = \frac{40!}{4! \cdot 36!} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,390.
  • The coefficient of xsx^s (which is qq) corresponds to (40s)\binom{40}{s}.
  • Here, s=36s = 36, thus:
    q=(4036)=(404)=91,390.q = \binom{40}{36} = \binom{40}{4} = 91,390.
  • Hence, we find:
    qp=91,39091,390=1.\frac{q}{p} = \frac{91,390}{91,390} = 1.
    Thus, the value of \frac{q}{p} is 1.

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