Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x < y < z$ and $x+y+z=12$ is

How many distinct combinations can be formed using one or more letters from the word $MISSISSIPPI$?

In how many ways can $35$ apples be distributed among $3$ boys such that each boy can have any number of apples?

In how many ways can a committee of $5$ members be formed from $6$ men and $4$ women such that the committee contains at least one woman?

There are two urns. Urn $A$ has $3$ distinct red balls and urn $B$ has $9$ distinct blue balls. From each urn,two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

The value of $r$ for which $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ^{20}C_{r-2} ^{20}C_2 + ... + ^{20}C_0 ^{20}C_r$ is maximum is