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

The digit in the unit place of the number $(183!) + (3^{183})$ is

The number of distinct primes dividing $12! + 13! + 14!$ is

All the letters of the word $TABLE$ are permuted and the strings of letters (may or may not have meaning) thus formed are arranged in dictionary order. Then the rank of the word $TABLE$ counted from the rank of the word $BLATE$ 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 number of natural numbers,between $212$ and $999$,such that the sum of their digits is $15$,is . . . . . .