k− nC1(k−1)+ nC2(k−2)− nC3(k−3)+⋯+(−1)n nCn(k−n)=

k− nC1(k−1)+ nC2(k−2)− nC3(k−3)+⋯+(−1)n nCn(k−n) =k− nC1k+ nC1+ nC2.k−2. nC2− nC3.k+3 nC3+⋯+(−1)n. nCn.k−(−1)n. nCn.n =k[ nC0− nC1+ nC2− nC3+⋯+(−1)n. nCn]+ nC1−2 nC2+3 nC3+⋯+(−1)n.n. nCn =k(0)+0=0 (1+x)n=C0+C1x+C2x2+⋯+Cnxn put x=−1 =C0−C1+C2−C3+⋯+(−1)n.Cn Differentiative n(1+x)n−1=C1+ 2C2.x+ 3C3x2+⋯nCnxn−1 put x=−1 0=C1−2C2+3.C3+⋯+(−1)nnCn

k− nC1(k−1)+ nC2(k−2)− nC3(k−3)+⋯+(−1)n nCn(k−n)=

k− nC1(k−1)+ nC2(k−2)− nC3(k−3)+⋯+(−1)n nCn(k−n) =k− nC1k+ nC1+ nC2.k−2. nC2− nC3.k+3 nC3+⋯+(−1)n. nCn.k−(−1)n. nCn.n =k[ nC0− nC1+ nC2− nC3+⋯+(−1)n. nCn]+ nC1−2 nC2+3 nC3+⋯+(−1)n.n. nCn =k(0)+0=0 (1+x)n=C0+C1x+C2x2+⋯+Cnxn put x=−1 =C0−C1+C2−C3+⋯+(−1)n.Cn Differentiative n(1+x)n−1=C1+ 2C2.x+ 3C3x2+⋯nCnxn−1 put x=−1 0=C1−2C2+3.C3+⋯+(−1)nnCn

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$k -^{n} C_{1} (k - 1) +^{n} C_{2} (k - 2) -^{n} C_{3} (k - 3) + \dots + {(- 1)}^{n}^{n} C_{n} (k - n) =$

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Detailed Solution

k -^{n} C_{1} (k - 1) +^{n} C_{2} (k - 2) -^{n} C_{3} (k - 3) + \dots + {(- 1)}^{n}^{n} C_{n} (k - n)

$= k -^{n} C_{1} k +^{n} C_{1} +^{n} C_{2} . k - 2.^{n} C_{2} -^{n} C_{3} . k + 3^{n} C_{3} + \dots + {(- 1)}^{n} .^{n} C_{n} . k - {(- 1)}^{n} .^{n} C_{n} . n$

$= k [^{n} C_{0} -^{n} C_{1} +^{n} C_{2} -^{n} C_{3} + \dots + {(- 1)}^{n} .^{n} C_{n}] +^{n} C_{1} - 2^{n} C_{2} + 3^{n} C_{3} + \dots + {(- 1)}^{n} . n .^{n} C_{n}$