Permutations an...

In how many ways atleast one horse and atleast one dog can be selected out of eight horses and seven dogs.

Determine $$n$$, if $$^ { 2 n } \mathrm { C } _ { 3 } : ^ { n } \mathrm { C } _ { 3 } = 12 : 1$$

The total number of ways of arranging the letters $$AAABBBCCDEF$$ in a row such that letters $$C$$ are separated from one another is

$$^{404}C_{4}-^{4}C_{1}^{303}C_{4}+^{4}C_{2}^{202}C_{4}-^{404}C_{4}^{101}C_{4}$$ is equal to

The number of positive integral solutions of the equation $$x _ { 1 } x _ { 2 } x _ { 3 } x _ { 4 } x _ { 5 } = 1050$$ is

Number of different natural numbers which are smaller than two hundred million $$\&$$ using only the digits $$1$$ or $$2$$ is ___________________.

If $$(1+x+x^2)^n=\displaystyle\sum^{2n}_{r=0}a_rx^r$$, then $$a_0a_{2r}-a_1a_{2r+1}+a_2a_{2r+2}-....=?$$

Number of cyphers at the end of 202$$\mathrm { C } _ { 1001 }$$ is ________.

If $$\displaystyle\sum^{n-r}_{k=1}$$ $$^{n-k}C_r={^{x}C_y}$$ then?

Letters of the word $$MATHEMATICS$$ are arranged in all the possible ways, in how many words letter $$C$$ is between $$S$$ and $$H$$(these three letter are not necessary together)?