Question

A man has $3$ friends. In how many ways he can invite one friend everyday for dinner on $6$ successive nights so that no friend is invited more than $3$ times

• A. $600$
• B. $510$
• C. $255$
• D. $592$

Answer 1

Answer: (B)

$510$

These are the possible ways

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 of them and we obtain the answer as $510$