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Question

# c0,c1,c2 denotes coefficents expansion of (1+x)n , then c1+c1c2+c2c3+.......cn−1cn=(2n)!(n+1)!(n−1)!

A
True
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B
False
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Solution

## The correct option is A TrueC0,C1,C2 denotes coefficients expansion of (1+x)n thenC1+C1C2+C2C3+…+Cn−1Cn=2n!(n−1)!(n+1)!we know that(1+x)n=C0+C1x+C2x2+C3x3+⋯+Cn−2xn−2+Cn−1xn−1+CnxnAlso we can write it as(1+x)n=Cnxn+Cn−1xn−1+⋯C2x2+C1x+C0Multiplying these two expressions(1+x)2n=(C0+C1x+C2x2+…+Cn−1xn−1+Cnxn)X(Cnxn+Cn−1xn−1+…+C2x2+C1x+C0)Now equate the coefficients ofxn−r from both sides of equationsWe get2nCn−r=C0Cn−r+C1Cn−r−1+C2Cn−r−2+C3Cn−r−3+…+Cn−rCnPut r=1C0C1+C1C2+C2C3+⋯+Cn−1Cn=2n!(n−1)!(n+1)!

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