An unbiased die is thrown 6 times. Calculate the probabilities that the six scores obtained will
i) be 1, 2,3, 4, 5,6 in some order
ii) have a product which is an even number
iii) consists of exactly two 6’s and four odd numbers
iii Alternative solution
6C2: 6 slots choose 2 slots to insert the two 6s. Remaining 4 slots automatically will be for the four odd numbers.
3^4: Each odd number has 3 choices. for 4 odd numbers will be 3^4
Probability = (6C2 x 3^4) / 6^6 = 5/192
Venn diagram, Inclusion and exclusion principle can be useful in solving permutations, combinations and probability problems.
It is especially useful in solving combinations where the cases are not mutually exclusive and hence addition principle cannot be applied.
A tennis club has n male players and n female players. For a tournament the players are to be arranged in n pairs, with each pair consisting of one male and one female. Find the number of possible pairings.
Suppose the male got to choose his partner. The first male can choose from n females. After he has chosen, the next male can choose from (n-1) females. So the number of possible pairings is n.(n-1).(n-2)…1 = n!
For instance there are 2 male players (M1, M2) and 2 female (F1, F2) players.
One way to arrange 2 pairs of players: M1 F1 and M2 F2
Second way to arrange 2 pairs of players: M1 F2 and M2 F1
Number of ways to arrange 2 pairs of players is 2!
Some students will give the wrong answer n x n, which is the number of ways to choose a pair.