The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples is

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples,$5$ white apples,and $8$ oranges. If in the selected $5$ fruits,at least $2$ oranges,at least $1$ red apple,and at least $1$ white apple must be included,then the number of ways Anil's mother can offer $5$ fruits to Anil is $........$

If $^nC_3 + ^nC_4 > ^{n+1}C_3$,then

Given $5$ different green toys,$4$ different blue toys,and $3$ different red toys,how many combinations of toys can be chosen taking at least one green and one blue toy?

In how many ways can we select any $4$ letters from the word $CORGOO$?

The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is