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 $(4 n +2), n \in N$ 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
$\therefore 1 \times 4 \times 2=8$
$\Rightarrow A , B , C$