Three married couples are to be seated in a row having six seats in cinema halls. If spouses are to be seated next to each other, in how many ways can they be seated? Also, find the number of ways of their seating, if all the ladies sit together.

If \(\frac { (2n)! }{ 3!(2n-3)! } \) and \(\frac { n! }{ 2!(n-2)! } \) are in the ration 44:3, find n

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
(i) repetition of the digits is allowed?
(ii) repetition of the digits is not allowed?

Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

\(find \ r \ if\quad ^{ 9 }P_{ 5 }+5.\quad ^{ 9 }P_{ 4 }=^{ 10 }P_{ r }\)

If \(\frac { n! }{ 5!(n-1)! } \ and\ \frac { n! }{ 7!(n-3)! } \) are in the ratio 21: 1, find the value of n.

If ²²P_r+1: ²⁰P_r+2 = 11:52, find r.

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Find if  \(^{ 5 }P_{ r }=^{ 6 }P_{ r-1 }\)

In how many ways can the letters of the word PERMUTATIONS be arranged if the
(i) words start with P and end with S
(ii) vowels are all together
(iii) there are always 4 letters between P and S?

How many numbers are there between 100 and 1000 such that at least one of their digits is 7?