The value of 30 C0 30 C15 215- 30 C1 29 C14 214+ 30 C2 28 C13 213 - ldots- 30 C15 15 C0 is

Question

The value of 30 C0 30 C15 215- 30 C1 29 C14 214+ 30 C2 28 C13 213 - ldots- 30 C15 15 C0 is (A) 0 (B)  30 C15 215 (C) - 30 C15 215 (D)  30 C15

Tardigrade Admin · Accepted Answer

S= 30 C0 30 C15 215- 30 C1 29 C14 214+ 30 C2 28 C13 213- ldots - 30 C15 15 C0 Tr= 30 Cr ⋅ 30-r C15-r(2) 15-r(-1)r =(30 !/(30-r) ! r !) ((30-r) !/(15-r) ! 15 !) 215-r(-1)r =(30 !/r !) (1/(15-r) ! 15 !) 215-r(-1)r =(30 !/15 ! 15 !) (15 !/r !(15-r) !) 215-r(-1)r = 30 C15 ⋅ 15 Cr(2)15-r(-1) r ∴ S= 30 C15(2-1)15= 30 C15

Q. The value of $^{30} C_{0}^{30} C_{15} 2^{15} -^{30} C_{1}^{29} C_{14} 2^{14} +^{30} C_{2}^{28} C_{13} 2^{13}$ $- \dots -^{30} C_{15}^{15} C_{0}$ is

$0$

$^{30} C_{15} 2^{15}$

$-^{30} C_{15} 2^{15}$

$^{30} C_{15}$

Q. The value of 30C0​30C15​215−30C1​29C14​214+30C2​28C13​213 −…−30C15​15C0​ is

0

30C15​215

−30C15​215

30C15​

Q. The value of $^{30} C_{0}^{30} C_{15} 2^{15} -^{30} C_{1}^{29} C_{14} 2^{14} +^{30} C_{2}^{28} C_{13} 2^{13}$ $- \dots -^{30} C_{15}^{15} C_{0}$ is

$0$

$^{30} C_{15} 2^{15}$

$-^{30} C_{15} 2^{15}$

$^{30} C_{15}$