Permutations an...

$$12$$ persons meet in a room and each shakes hands with all the others. How many handshakes are there?

Out of $$5$$ men and $$2$$ women, a committee of $$3$$ is to be formed. In how many ways can it be formed if at least one woman is included in each committee?

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

The exponent of 3 in $$100!$$ is 

The total number of 5- digit telephone numbers that can be composed with distinct digits, is

The letters of word 'ZENITH' are written in all positive ways. If all these words are written in the order of a dictionary, then the rank of the word 'ZENITH' is

If $$^{n}C_{3} + ^{n}C_{4} > ^{n + 1}C_{3}$$, then

Find the values of $$ ^{61}C_{57} -^{60}C_{56} $$

If $$ ^{15}C_{3r} = ^{15}C_{r+3} $$ then r equal to :

$$ ^{ 47 }C_{ 4 }$$ +$$ \sum _{ r=1 }^{ 5 }$$ .$$^{ 52-r }C_{ 3 } $$is equal to :