Permutations

First we fix the alternate position of the girls.
Five girls can be seated around the circle in boys can be seated in five -vacant place by
Required number of ways

A combination of four vertices is equivalent to one interior points of intersection of diagonals.
No. of interior points of intersection



So, number of diagonals

First prize may be given to any one of the boys, hence first prize can be distributed in ways.
similarly every one of second, third, fourth and fifth prizes can also be given in ways.
the number of ways of their distribution

We have : .
So, can be assigned to either or or
i.e. can be assigned in ways.
Similarly, each of and can be assigned in ways.
Thus, the number of solutions is .

A committee of out of can be
made in ways.
If no woman is to be included,
then number of ways = 6
the required number

No. of words starting with are
No. of words starting with are
No. of words starting with are
These account for words
Next word is RAHLU and the word RAHUL.

The four digits can be arranged at places in ways.
The five digits can be arranged at place in ways.
Total number of arrangements is .

First prize can be given in ways. Then second prize can be given in ways and the third prize in ways (Since a competitior cannot get two prizes) and hence the no. of ways. = ways

Digits available are . The 4-digit number must be divisible by 5. A number is divisible by 5 if its last digit is 0 or 5. Since 0 is not available, the last digit must be 5. Step 1: Fix the last digit as 5. Step 2: The first three digits are to be chosen from without repetition. Step 3: Number of ways to arrange the first three digits = Step 4: Total number of such 4-digit numbers = .