SOLUTION: What is the co-efficient of {{{ x^10 }}} in the following expressions:
(A) 1 + (1 + x) + (1 + x)^2 + ... + (1 + x)^20
(B) (1 + x)^2 + (1 + x)^3 + (1 + x)^4 + (1 + x)^5 + .... + (
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Question 1184312: What is the co-efficient of in the following expressions:
(A) 1 + (1 + x) + (1 + x)^2 + ... + (1 + x)^20
(B) (1 + x)^2 + (1 + x)^3 + (1 + x)^4 + (1 + x)^5 + .... + (1 + x)^10
[Note: Here ^ means power]
Found 2 solutions by robertb, greenestamps:
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
(A)
=
=
(B) This should be more manageable than (A). Note that the term appears in the last term of the expression only, which is .
Therefore the coefficient of is
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
The second one is trivial; only the expansion of the last term in the sum contains an x^10 term, and the coefficient of that term is clearly 1.
(B) ANSWER: 1
In the first one, there is no x^10 term until we get to (1+x)^10. Then the coefficients of the x^10 terms in the expansion of the remaining terms in the given sum are...
(1+x)^10: C(10,10)
(1+x)^11: C(11,10)
(1+x)^12: C(12,10)
...
(1+x)^20: C(20,10)
The coefficient of the x^10 term in the sum is then found using the "hockey stick" equation for Pascal's Triangle:
C(10,10)+C(11,10)+C(12,10)+...+C(20,10) = C(21,11)
ANSWER: C(21,11) (which is the same as C(21,10))
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