Question

Let the number of ways to invite one of the three friends for dinner on 6 successive nights such that no friend is invited more than 3 times be $K$. Find sum of digits of $K$ ?

Answer 1

These are the possible ways you choose to invite your friends and the total count

You can invite any two friends three nights and this you can do three number of ways

$3{\times }_3^{6}C{\times }_3^{3}C=60$

You can invite any friend three times, second friend 2 times and third friend one time. This you can do 6 number of ways.

$6{\times }_3^{6}C{\times }_2^{3}C{\times }_1^{1}C=360$

You can invite 1st friend two times, second friend two times and third friend two times. This you can do 1 number of ways:

${}_2^{6}C{\times }_2^{4}C{\times }_2^{2}C=90$

Adding all we get total possibility $60+360+90=510$

Sum of digits $=5+1+0=6$