Let $A$ be a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen, and the set $A$ is reconstructed by replacing the elements of $P$. $A$ subset $Q$ of $A$ is chosen again. The number of ways of choosing $P$ and $Q$ such that $Q$ contains just one element more than $P$ is

If for some $m, n$,${ }^6 C_{m}+2({ }^6 C_{m+1})+{ }^6 C_{m+2} > { }^8 C_3$ and ${ }^{n-1} P_3 : { }^n P_4 = 1 : 8$,then ${ }^n P_{m+1} + { }^{n+1} C_m$ is equal to

The total number of $5$-digit telephone numbers that can be formed such that at least one digit is repeated is .... (The first digit cannot be $0$).

There are $(n + 1)$ white balls and $(n + 1)$ black balls. Each ball is numbered from $1$ to $(n + 1)$. In how many ways can these balls be arranged in a row such that no two balls of the same color are adjacent?

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to $.......$.

Three and four digit numbers are formed from the digits $1, 3, 5, 6, 8$. If $e_1$ is the number of three-digit even numbers with no digit repeated and $e_2$ is the number of four-digit even numbers with no digit repeated. Also,$O_1$ represents the number of three-digit odd numbers in which no digit is repeated and $O_2$ represents the number of four-digit odd numbers in which no digit is repeated. Then: