SOLUTION: a) Simplify \[\frac{\binom{n}{k}}{\binom{n}{k - 1}}.\] B) For some positive integer n, the expansion of (1 + x)^n has three consecutive coefficients a,b,c that satisfy a:b:c =

Algebra ->  Statistics  -> Binomial-probability -> SOLUTION: a) Simplify \[\frac{\binom{n}{k}}{\binom{n}{k - 1}}.\] B) For some positive integer n, the expansion of (1 + x)^n has three consecutive coefficients a,b,c that satisfy a:b:c =       Log On


   



Question 1164006: a) Simplify
\[\frac{\binom{n}{k}}{\binom{n}{k - 1}}.\]
B) For some positive integer n, the expansion of (1 + x)^n has three consecutive coefficients a,b,c that satisfy a:b:c = 1:7:35. What must n be?

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


part (a)....

C%28n%2Ck%29+=+%28n%21%29%2F%28%28k%21%29%28n-k%29%21%29
C%28n%2Ck-1%29+=+%28n%21%29%2F%28%28%28k-1%29%21%29%28n-k%2B1%29%21%29



part (b)....

C%28n%2Ck%29%2FC%28n%2Ck-1%29+=+%28n-k%2B1%29%2Fk

C%28n%2Ck-1%29%2FC%28n%2Ck-2%29+=+%28n-k%29%2F%28k%2B1%29

The problem tells us

C%28n%2Ck%29%2FC%28n%2Ck-1%29+=+7%2F1
C%28n%2Ck-1%29%2FC%28n%2Ck-2%29+=+35%2F7+=+5%2F1

So

%28n-k%2B1%29%2Fk+=+7
%28n-k%29%2F%28k%2B1%29+=+5

Solve the pair of equations for k and n:

7k+=+n-k%2B1
5%28k%2B1%29+=+n-k

8k-n+=+1
6k-n+=+-5
2k+=+6
k+=+3
n+=+23

ANSWER: n = 23 (and k=3)

CHECK:
C%2823%2C3%29%2FC%2823%2C2%29+=+1771%2F253+=+7
C%2823%2C4%29%2FC%2823%2C3%29+=+8855%2F1771+=+5