Resolve into partial factors:\(\frac { x+4 }{ ({ x }^{ 2 }-4)(x+1) } \)

Solve : \(\frac { (2x+1)! }{ (x+2)! } .\frac { (x-1)! }{ (2x-1)! } =\frac { 3 }{ 5 } \)

How may different numbers between 100 and 1000 can be formed using the digits 0, 1,2,3,4, 5, 6 assuming that in any number, the digits are not repeated.

There are 6 gentlemen and 4 ladies to line at a round table. In how many ways can they seat themselves so that no two ladies together?

In how many ways can n prizes be given to n boys, when a boy may receive any number of prizes?

In an examination, Yamini has to select 4 questions from each part. There are 6, 7 and 8 questions is Part I, Part II and Part III respectively. What is the number of possible combinations in which she can choose the questions?

In how many ways can 12 things be equally divided among 4 persons?

If p(n) is the statement "12n + 3" is a multiple of 5, then show that P (3) is false, whereas P(6) is true.

Let p(n) be the statement "n² + n is even". If P(k) is true, then show that P(k+1) is true.

If tan \(\alpha={{1}\over{7}},\sin\beta{{1}\over{\sqrt{10}}},\) Prove that \(\alpha+2\beta{{\pi}\over4{}}\) where \(0<\alpha<{{\pi}\over{2}}\) and \(0<\beta<{{\pi}\over{}2}.\)