In how many ways can $mn$ letters be posted in $n$ letter-boxes?

The number of ways in which the letters of the word $TRIANGLE$ can be arranged such that no two vowels occur together is

Statement-$1$: The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3$.
Statement-$2$: The number of ways of choosing any $3$ places from $9$ different places is $^9C_3$.

The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ in which at least two digits are identical is

How many five-digit numbers can be formed using the digits $2, 0, 4, 3, 8$ if repetition of digits is not allowed?

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves exceeded the number of games that the men played with the women by $66$. The total number of participants is: