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?

If four letters are chosen from the letters of the word $ASSIGNMENT$ and are arranged in all possible ways to form $4$-letter words (with or without meaning),then the total number of such words that can be formed is:

The number of $7$-$digit$ numbers which are multiples of $11$ and are formed using all the digits $1, 2, 3, 4, 5, 7,$ and $9$ is

Four-digit numbers with all digits distinct are formed using the digits $1, 2, 3, 4, 5, 6, 7$ in all possible ways. If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than $3400$ among them,then $p: q=$

If $_nP_r = 30240$ and $\binom{n}{r} = 252$,then $(n, r) = \dots$

$A$ person forgets his $4-$digit $ATM$ pin code. But he remembers that in the code all the digits are different,the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is $...........$.