The value of 30 C0- 30 C1+ 30 C2- 30 C3+ ldots+ 30 C10 is

Question

The value of 30 C0- 30 C1+ 30 C2- 30 C3+ ldots+ 30 C10 is (A)  29 C10 (B)  29 C9 (C) - 29 C10 (D) - 29 C20

Tardigrade Admin · Accepted Answer

30 C0 - 30 C1+ 30 C2- 30 C3+ ldots+ 30 C10 = Coefficient of x10 in (1-x)30(1+x+x2+ ldots+x10+ ldots) = Coefficient of x10 in (1-x)30(1-x)-1 =Coefficient of x10 in (1-x)29 = 29 C10

Q. The value of $^{30} C_{0} -^{30} C_{1} +^{30} C_{2} -^{30} C_{3} + \dots +^{30} C_{10}$ is

$^{29} C_{10}$

$^{29} C_{9}$

$-^{29} C_{10}$

$-^{29} C_{20}$

$^{30} C_{0} -^{30} C_{1} +^{30} C_{2} -^{30} C_{3} + \dots +^{30} C_{10}$
= Coefficient of $x^{10}$ in $(1 - x)^{30} (1 + x + x^{2} + \dots + x^{10} + \dots)$
= Coefficient of $x^{10}$ in $(1 - x)^{30} (1 - x)^{- 1}$
=Coefficient of $x^{10}$ in $(1 - x)^{29}$
$=^{29} C_{10}$

Q. The value of 30C0​−30C1​+30C2​−30C3​+…+30C10​ is

29C10​

29C9​

−29C10​

−29C20​