Permutations an...

A shelf contains $$15$$ books, of which $$4$$ are single volume and the others are $$8$$ and $$3$$ volumes respectively. In how many ways can these books be arranged on the shelf so that order of the volumes of same work is maintained $$?$$

$$\displaystyle \sum^{n-1}_{r=0}\dfrac {^{n}C_{r}}{^{n}C_{r}+^{n}C_{r+1}}=$$

If  $$a=\,^ { m }C _ { 2 } ,$$  then  $$^ { a } C _ { 2 }$$ is equal to

$$^{ n }{ C }_{ 1 }.2+^{ n }{ C }_{ 2 }.\frac { { 2 }^{ 2 } }{ 3 } +^{ n }{ C }_{ 3 }.\frac { { 2 }^{ 3 } }{ { 3 }^{ 2 } } +......^{ n }{ C }_{ n }.\frac { { 2 }^{ n } }{ { 3 }^{ n-1 } } =$$

The number of seven letter words that can be formed by using the letters of the word  $$SUCCESS$$  that the two  $$C$$ are together but no two  $$S$$  are together is

Nine boys and 3 girls are to be seated in 2 vans, each having numbered seats, 3 in front and 4 at back. The number of ways of seating arrangements, if the girls should sit together in a back row on adjacent seats, is 

Six people are going to sit in a row on a bench. $$A$$ and $$B$$ are adjacent, $$C$$ does not want to sit adjacent to $$D.E$$ and $$F$$ can sit anywhere. Number of ways in which these six people can be seated is 

How many different words can be formed by jumbling the letters in the word  $$MISSISSIPPI$$  in which no two  $$S$$  are adjacent ?

In the expansion of $$\left(x^3 - \dfrac{1}{x^2}\right)^{15}$$, the constant terms is

The value of $$^{47}C_{4}+\displaystyle \sum _{ j=1 }^{ 5 }\ ^{ \left( 52-j \right)  } { C }_{ 3 }$$ is