Permutations an...

$$\displaystyle \sum _{ 0\le i\le  }^{  }{ \sum _{ j\le 10 }^{  }{ ^{ 10 }{ { C }_{ j } }^{ j }{ { C }_{ i } } }  } $$ is equal to

The sum $$\displaystyle \sum_{i=0}^{m}\binom{10}{i}\binom{20}{m-i}$$ be maximum when m is

The value of the expression 
$$\displaystyle2^{k}\binom{n}{0}\binom{n}{k}-2^{k-1}\binom{n}{1}\binom{n-1}{k-1}+2^{k-2}\binom{n}{2}\binom{n-2}{k-2}..+(-1)^{k}\binom{n}{k}\binom{n-k}{0}$$ is

If $$\displaystyle ^{100}C_{3} = 161700$$, then $$^{100}C_{97}$$ is equal to___.

A road network as shown in the figure connect four cities. In how many ways can you start from any city (say A) and come back to it without travelling on the same road more than once ?

If $$\displaystyle ^{n}C_{3}=  ^{n}C_{5'}$$ then find the value of n:

If $$ { _{  }^{ n }{ C } }_{ 4 },{ _{  }^{ n }{ C } }_{ 5 }$$ and $$ { _{  }^{ n }{ C } }_{ 6 }$$ are in AP, then $$n$$ is

If $$^{19}C_r$$ and $$^{19}C_{r-1}$$ are in the ratio 2:3, then find $$^{14}C_r$$

A bag contains n white and n black balls. Pairs of balls are drawn at random without replacement successively, until the bag is empty. If the number of ways in which each pair consists of one white and one black ball is 14,400. then n =

Three persons entered a railway compartment in which $$5$$ seats were vacant. Find the number of ways in which they can be seated