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Question
How many of the following statements are true?
(1) rnCr=n×n−1Cr−1
(2) Each term of the sum
(2nC1)2+2×(2nC2)2+3.(2nC3)2.....+2n.(2nC2n)2 can be written as r(2nCr)2
(3) coefficient of x2n−1 in
(2n−1C0+2n−1C1x+2n−1C2x2.....2n−1C2n−1x2n−1)×(2nC0+2nC1x+2nC2x2.....2nC2nx2n)
= 2n−1C0×2nC2n−1+2n−1C1×2nC2n−2.....2n−1C2n−1.2nC0
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How many of the following statements are true?
(1) rnCr=n×n−1Cr−1
(2) Each term of the sum
(2nC1)2+2×(2nC2)2+3.(2nC3)2.....+2n.(2nC2n)2 can be written as r(2nCr)2
(3) coefficient of x2n−1 in
(2n−1C0+2n−1C1x+2n−1C2x2.....2n−1C2n−1x2n−1)×(2nC0+2nC1x+2nC2x2.....2nC2nx2n)
= 2n−1C0×2nC2n−1+2n−1C1×2nC2n−2.....2n−1C2n−1.2nC0
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Solution
(1)nCr=n!(n−r)!×r!
=n×(n−1)!r(n−r)!×(r−1)!
=nr×n−1Cr−1
(2)Lets say each term is r(2nCr)2. When we put r=0,1,2,3.... we get 0,(2nC1)2,2(2nC2)2,3(2nC1)2.....Which are the terms of the given sum.
(3)In the expansion of
(2n−1C0+2n−1C1x+....2n−1C2n−1x2n−1)×(2nC0+2nC1x+2nC2x2.....),
We get the terms with x2n−1 when we multiply
(2n−1C0 and 2nC2n−1x2n−1),
(2n−1C1x and 2nC2n−2x2n−2),....... and
(2n−1C2n−1X2n−1 and 2nC0),
(1)nCr=n!(n−r)!×r!
=n×(n−1)!r(n−r)!×(r−1)!
=nr×n−1Cr−1
(2)Lets say each term is r(2nCr)2. When we put r=0,1,2,3.... we get 0,(2nC1)2,2(2nC2)2,3(2nC1)2.....Which are the terms of the given sum.
(3)In the expansion of
(2n−1C0+2n−1C1x+....2n−1C2n−1x2n−1)×(2nC0+2nC1x+2nC2x2.....),
We get the terms with x2n−1 when we multiply
(2n−1C0 and 2nC2n−1x2n−1),
(2n−1C1x and 2nC2n−2x2n−2),....... and
(2n−1C2n−1X2n−1 and 2nC0),
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