1. A coin is tossed 3 times and the outcomes are record .How many possible outcomes are these ?

 When a coin is to tossed once,the number of outcomes is 2.ie ,In each throw the number of ways of showing a different face is 2.
Thus ,the required possible outcomes is 2×2×2=8

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

 Each signal required the use of 2 flags.There will be as many flags as there are ways of filling in 2 vacant plates   in succession by the given 5 flags of different colours. The upper vacant place can be filled in 5 different ways by any one of the 5 flags following which the lower vacant place can be filled in 4 different ways by any of the remaining 4 different flags.Thus ,by multiplication principle,the number of different signals that can be generated is 5×4=20

3. Evaluate

(i) 8! (ii) 4!-3!

(i) 8! = 1×2×3×4×5×6×7×8 = 40320
(ii) 4! =1×2×3×4 = 24
          3! = 1×2×3 = 6
∴ 4! - 3! = 24 - 6 = 18

4. Compute ^8!/6!×2!

^8!/6!×2! = ^8×7×6! / 6!×2×1 = ^8×7/2 = 28

5. Evaluate ^n! / (n-r)!, when n=6,r=2.

When n = 6 , r = 2 , ^n! /(n-r)! = ^6!/(6-2)! = ^6!/4!
                                               = ^6×5×4!/4! = 30

6. How many 4-digit numbers are there with no digit repeated ?

The thousands place of the 4-digit number is to be filled with any of the digits from 1 to 9 as the digit 0 cannot be included.
∴ The number of ways in which thousands places can be filled is 9.
 The hundreds ,tens,and units place can be filled by any of the digits from o to 9 .However ,the digits cannot be repeated in the 4-digit numbers and thousands place is already occupied with a digit.The hundred ,tens and units place is to be filled by the remaining 9 digits.
∴ There will be as many such 3-digit number as there are permutations of 9 different digits taken 3 at a time .
Number of such 3-digit numbers
= 9p₃ = ^9!/(9-3)! = ^9!/6!
            = ^9×8×7×6!/6! = 9×8×7 = 504
Thus ,by multiplication principle ,the required number of 4-digit numbers is 9×504 = 4536.