Question

Statement-1: The number `1000C_(500)` is not divisible by 11.
Statement-2: The exponent of prime p in n! is
`[(n)/(p)]+[(n)/(p^(2))]+[(n)/(p^(3))]+......+[(n)/(p^(x))]" where "p^(k)lenltp^(k+1)`
A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Statement-1 is true (see section 2 on page 13.2) We have,
`""^(1000)C_(500)=(1000!)/(500!500!)`
`E_(11)(1000!)=[(1000)/(11)]+[(1000)/(11^(2))]=90+8=98`
`E_(11)=[(500)/(11)]+[(500)/11^(2)]=45+4=49`
Hence, `""^(1000)C_(500)` is not divisible by 11.
Thus, both statements aer true and statement-2 is a correct explanation for statement-1.