SOLUTION: In the expansion of (1+x)^21,the coefficient of (2r+1)th term is equal to the coefficient of (3r+2)th term, then find the value of r.

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Question 1089122: In the expansion of (1+x)^21,the coefficient of (2r+1)th term is equal to the coefficient of (3r+2)th term, then find the value of r.
Answer by KMST(5328) About Me  (Show Source):
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The expansion of a binomial %28a%2Bb%29%5En has n%2B1 terms.
They are of the form Ca%5Epb%5Eq with a combinatorial coefficient C ,
and positive integers p and q such that p%2Bq=n .
There's is a certain symmetry to the combinatorial coefficients:
The first and last ones (the number 1 and n%2B1 coefficients are 1;
the number 2 and n coefficients are n ;
the number 3 and n-1 coefficients are n%28n-1%29%2F2 ,
and so on, so that the number m and s coefficients will be the same if m%2Bs=n%2B2 .
So, in this case,
%282r%2B1%29%2B%283r%2B2%29=21%2B2 ,
5r%2B3=23 ,
5r=20 ,
and highlight%28r=4%29 .

Verification:
2r%2B1=2%2A4%2B1=8%2B1=9
3r%2B2=3%2A4%2B2=12%2B2=14
The 9th coefficient is
.
The 14th coefficient is
.