Q. Consider the following statements : 1.The product of $r$ consecutive natural numbers is always divisible by $r$. 2.The total number of proper positive divisors of 115500 is 94 3. A pack of 52 cards can be divided equally among four players order in $\frac{52!}{(13!)^{4}}$ ways. Which of the statement given above is/are correct?
A) Only (1)
B) Only (2)
C) Only (3)
D) All of (1), (2) and (3)
Explanation
(d)
(1)It is true that product of $r$ consecutive natural numbers is always divisible by $r$.
(2) Now, 115500=$2^{2} \times 3^{1} \times 5^{3} \times 7^{1} \times 11^{1}$
$\therefore$ Total number of proper divisor
$= (2 + 1)(1 + 1)(3 + 1)(1 + 1)(1 + 1) - 2$
$= 96 - 2 = 94$
(3) Total number of ways $= \frac{52!}{(13!)^{4}}$
Hence, all statements are true
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.