Here you will learn some permutation and combination examples for better understanding of permutation and combination concepts.
Example 1 : If all the letters of the word ‘RAPID’ are arranged in all possible manner as they are in a dictionary, then find the rank of the word ‘RAPID’.
Solution : First of all, arrange all letters of given word alphabetically : ‘ADIPR’
Total number of words starting with A _ _ _ _ = 4! = 24
Total number of words starting with D _ _ _ _ = 4! = 24
Total number of words starting with I _ _ _ _ = 4! = 24
Total number of words starting with P _ _ _ _ = 4! = 24
Total number of words starting with R A D _ _ = 2! = 2
Total number of words starting with R A I _ _ = 2! = 2
Total number of words starting with R A P D _ = 1 = 1
Total number of words starting with R A P I _ = 1 = 1
\(\therefore\) Rank of the word RAPID = 24 + 24 + 24 + 24 + 2 + 2 + 1 + 1 = 102
Example 2 : Find the number of ways of dividing 52 cards among 4 players equally such that each gets exactly one Ace.
Solution : Total number of ways of dividing 48 cards(Excluding 4 Aces) in 4 groups = \(48!\over (12!)^4 4!\)
Now, distribute exactly one Ace to each group of 12 cards. Total number of ways = \(48!\over (12!)^4 4!\) \(\times\) 4!
Now, distribute these groups of cards among four players
= \(48!\over (12!)^4 4!\) \(\times\) 4!4! = = \(48!\over (12!)^4\) \(\times\) 4!
Example 3 : How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
Solution : There are 4 odd digits (1, 1, 3, 3) and 4 odd places(first, third, fifth and seventh). At these places the odd digits can be arranged in \(4!\over 2!2!\) = 6
Then at the remaining 3 places, the remaining three digits(2, 2, 4) can be arranged in \(3!\over 2!\) = 3 ways
\(\therefore\) The required number of numbers = 6 \(\times\) 3 = 18
Example 4 : In how many ways can 5 different mangoed, 4 different oranges & 3 different apples be distributed among 3 children such that each gets atleast one mango?
Solution : 5 different mangoes can be distributed by following ways among 3 children such that each gets at least 1 :
Total number of ways : (\(5!\over 3!1!1!2!\) + \(5!\over 2!2!2!\)) \(\times\) 3!
Now, the number of ways of distributing remaining fruits (i.e. 4 oranges + 3 apples) among 3 children = \(3^7\) (as each fruit has 3 options).
\(\therefore\) Total number of ways = (\(5!\over 3!2!\) + \(5!\over {(2!)}^3\)) \(\times\) 3! \(\times\) \(3^7\)
Practice these given permutation and combination examples to test your knowledge on concepts of permutation and combination.
