How many different necklaces can be formed with $6$ white and $5$ red beads?

How many different words can be formed by jumbling the letters in the word $MISSISSIPPI$ in which no two $S$ are adjacent?

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

If $\sum_{i=0}^4 {^{4+i}}C_i + \sum_{j=6}^9 {^{3+j}}C_j = {^x}C_y$ ($x$ is a prime number),then which one of the following is incorrect?

In how many ways can a committee,consisting of $5$ men and $6$ women,be formed from $8$ men and $10$ women?

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