R S Aggarwal Solutions for Exercise 5: EXERCISE 5E

Find the number of ways in which $5$ ladies and $5$ gentlemen may be seated in a row so that no two ladies are together.

Find the number of ways in which $m$ boys and $n$ girls may be arranged in a row so that no two of the girls are together, it being given that $m>n$.

In how many ways can $5$ children be arranged in a line such that two of them, Ram and Shyam, are always together?

In how many ways can $5$ children be arranged in a line such that two of them, Ram and Shyam, are never together?

When a group photograph is taken, all the seven teachers should be in the first row and all the twenty students should be in the second row. If the two corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal, how many arrangements are possible?

Find a formula for the number of permutations of $n$ different things taken $r$ at a time such that two specified things occur together.

How many numbers divisible by $5$ and lying between $3000$ and $4000$ can be formed by using the digits $3, 4, 5, 6, 7, 8$ when no digit is repeated in any such number?

In an examination $10$ candidates have to appear. $4$ candidates are to appear in mathematics and the rest in different subjects. In how many ways can they be seated in a row, if candidates appearing in mathematics are not to sit together?