Question

A man has 3 friends. Number of ways in which he can invite one friend every day for dinner on 6 successive nights so that he invites every friend atleast once is N. Which of the following hold(s) good?

• A. Number of divisors of $N$ which are divisible by 5 is 12 .
• B. Sum of all the divisors of $N$ is 1680 .
• C. Number of ways in which $N$ can be resolved as product of two divisors which are relatively prime, is 4 .
• D. Number of divisors of the form $(4n+2),n\in N$ is 6 .

Answer 1

Answer: (A), (B), (C)

$x,y,z$ friend
a times, b times, c times
$a,b,c\equiv (1,1,4),(1,2,3),(2,2,2)$
$\text{ Number of ways }=\frac{6!3!}{4!1!1!2!}+\frac{6!3!}{1!2!3!}+\frac{6!3!}{2!2!2!3!}=90+360+90=540$
$N\equiv 540=2^2\cdot 3^3\cdot 5^1$
(A) divisors divisible by 5 is: $3\times 4\times 1=12$
(B) Sum $=\left(2^0+2^1+2^2\right)\left(3^0+3^1+3^2+3^3\right)\left(5^0+5^1\right)$
$=7\times 40\times 6=1680$
(C) $2^{n-1}=2^2=4$
(D) Even and not divisible by 4
$\therefore 1\times 4\times 2=8$
$\Rightarrow A,B,C$