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(a) Simplify \(\frac{\binom{n}{k}}{\binom{n}{k - 1}}.\)

 

(b) In the expansion of (1 + x)n, there exist three consecutive coefficients a,b,c that satisfy a:b:c = 1:7:35. Find the positive integer n.

 Feb 2, 2020
edited by Guest  Feb 2, 2020
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Neat question.

(a+b) n can be written as

Sum[k={0,n} Choose(n,k) a kb n-k]

i.e. Choose(n,0) b n+Choose(n,1) a b n-1 + ... Choose(n,n-1) a n-1b + Choose(n,n) a n

Now you're probably thinking ok wth is "Choose"?

This is the binomial coefficient, "n choose k" and they are the values of a given row in Pascal's triangle. Come down n rows and go over k elements in the triangle and you get the Binomial coefficient corresponding to "n choose k" = Choose(n,k) = Binomial(n,k) (different names for the same thing).

So in this case if you look at the n=5 row (really the 6th row, the first row is n=0) of Pascal's triangle you see it's

1 5 10 10 5 1

in other words

Choose(5,0)=Choose(5,5) = 1
Choose(5,1)=Choose(5,4) = 5
Choose(5,2)=Choose(5,3) = 10

so (a+b) 5 = 1*a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 a b 4 + 1 b 5

now let a=3t and b=4 and substitute those into the formula above and simplify things and you're done

 Feb 2, 2020

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