Permutations an...

$$\displaystyle \begin{pmatrix} 47 \\ 4 \end{pmatrix}$$+$$\displaystyle \sum_{j=1}^{5}$$ $$\displaystyle \begin{pmatrix} 52-j \\ 3 \end{pmatrix}$$=$$\displaystyle \begin{pmatrix} x \\ y \end{pmatrix}$$ then $$\displaystyle \frac{x}{y}$$ =

The mean of the values $$0, 1, 2, \cdots n$$, having corresponding weights $$^nC_0, ^nC_1, \cdots ^nC_n,$$ respectively is

An $$n-$$ digit number is a positive number with exactly $$n$$ digits. Nine hundred distinct n−digit numbers are to be formed using only the three digits 2,5 and 7. The smallest value of $$n$$ for which this is possible is

If $$^{20}C_{r\, +\, 2}\, =\, ^{20}C_{2r\, -\, 3}\,$$, find  $$\, ^{12}C_r$$.

The number of such numbers which are divisible by two and five (all digits are not different) is

A set contains $$(2n + 1)$$ elements. The number of subsets of the set which contain at most n elements is

If $$'n'$$ is an integer between $$0$$ and $$21$$, then the minimum value of $$n!\left( 21-n \right) !$$ is

If $$^8C_r = ^8C_{r+2}$$, then the value of $$^rC_2$$ is:

$$^{50}C_{11}+^{50}C_{12}+^{51}C_{13}-^{52}C_{13}=$$

The value of  $$^{95}C_4+\displaystyle \sum_{j=1}^5 {\;}^{100-j}C_3$$ is