# A person wishes to make as many different parties as he can out of 10 of his friends. Each party consist of same no. of friends. Out of these parties in how many parties a particular friend will be found?

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• $^nC_r$ is maximum if $r=\large\frac{n}{2}$ or $r=\large\frac{n+1}{2}$
Let him invite $r$ friends at a time in each party.
Then the number of parties = $^{10}C_r$
This is maximum only if $r=5$
Out of these $^{10}C_5$ parties a particular person can be found in
$^9C_4 =126$ parties.
This is because out of any 5 friends 1 particular friend is included and the
remaining 4 friends are to be selected from remaining 9 friends.=$^9C_4$ (ways.)