$^nC_r + ^{n-1}C_r + ...... + ^rC_r$ =

How many numbers lying between $500$ and $600$ can be formed with the help of the digits $1, 2, 3, 4, 5, 6$ when the digits are not to be repeated?

How many numbers of $6$ digits can be formed from the digits of the number $112233$?

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is

The sum of all three-digit numbers,formed using non-zero digits,where all the digits are perfect squares of a natural number,is:

How many odd numbers less than $1000$ can be formed using the digits $0, 2, 5, 7$? (repetition of digits is allowed).