Monday, July 17th, 2017
As an astute man Mr. Gump once said, “Life is like a box of chocolates. You never know what you going to get.” The Permutations and Combinations that life presents us daily is baffling and probably it is because of that inherent fear of choices and cases we get intimidated by such questions in the exam. I understand that P & C is one of the dreaded topics but I hope that once you understand the fundas given below, your fear will reduce.
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Funda 1: De-arrangement
If ‘n’ distinct items are arranged in a row, then the number of ways they can be rearranged such that none of them occupies its original position is:
Note: De-arrangement of 1 object is not possible.
Dearr(2) = 1; Dearr(3) = 2; Dearr(4) =12 – 4 + 1 = 9; Dearr(5) = 60 – 20 + 5 – 1 = 44
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Eg1.1: A person has eight letters and eight addressed envelopes corresponding to those letters. In how many ways can he put the letters in the envelopes such that exactly 5 of them get delivered correctly?
Solution: At first, select the five letters that get delivered correctly. That can be done in ^{8}C_{5} ways.
Now, the other three must get delivered to the wrong address. That can be done in Dearr(3) = 2 ways.
So, total ways is 2 x ^{8}C_{5} = 2 x 56 = 112 ways.
Funda 2: Partitioning
‘n’ identical items in ‘r’ distinct groups | No restrictions: ^{n+r-1}C_{r-1} |
No group empty: ^{n-1}C_{r-1} | |
‘n’ distinct objects in ‘r’ distinct groups | No restrictions: r^{n} |
Arrangement in a group is important: |
Note: Other than standard distribution / partitioning problems, these ideas can be used to solve questions in which number of solutions are asked.
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Eg 2.1: How many solutions are there to the equation a + b + c = 100; given that
a) a, b and c are whole numbers.
b) a, b and c are natural numbers.
Solution:
Case a) is identical to a case in which 100 identical chocolates are being distributed in three kids a, b and c. It is possible that one kid gets all the chocolates. In this case, we will use the formula for distributing ‘n’ identical items in ‘r’ distinct groups where n = 100 and r = 3.
So, it can be done in ^{102}C_{2} ways.
Case b) is identical to a case in which 100 identical chocolates are being distributed in three kids a, b and c. Every kid must get at least one chocolate. In this case, we will use the formula for distributing ‘n’ identical items in ‘r’ distinct groups where no group is empty and n = 100 and r = 3.
So, it can be done in ^{99}C_{2} ways.
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Eg 2.2: In how many ways can you distribute 5 rings in
a) 4 boxes.
b) 4 fingers.
Solution: First of all we need to identify the difference between distributing in boxes and distributing in 4 fingers. The distinction is that in case of fingers, unlike boxes, the order in which rings are placed matters.
In Case a; Ring 1 can go in any of the four boxes, so it has four choices. Ring 2 can also go in any of the four boxes, so it has four choices. Similarly for Ring 3, Ring 4 and Ring 5; there are 4 choices each. So, the total number of ways of distribution is = 4 x 4 x 4 x 4 x 4 = 4^{5}. This is essentially how the formula r^{n} is derived.
In Case b; Ring 1 can go in any of the four fingers, so it has 4 choices.
Ring 2 can go in any of the four fingers but it has five choices. There is a finger, say F3, which contains the ring R1. Now, on F3, R2 has two choices – it can go above R1 or below R1. So, the total number of choices for R2 is 5.
Ring 3 can go in any of the four fingers but it now has 6 choices.
Ring 4 can go in any of the four fingers but it will now have 7 choices.
Ring 5 can go in any of the four fingers but it will now have 8 choices.
So, the total number of way of distribution of rings is =
This is essentially how the
Funda 3:
Number of ways of arranging ‘n’ items, out of which ‘p’ are alike, ‘q’ are alike and ‘r’ are alike given that p + q + r = n
Number of ways of distributing ‘n’ distinct items, in groups of size ‘p’, ‘q’ and ‘r’ given that p + q + r = n
I hope that this would help you solve problems in the exam. May be the chocolate you end up getting is a Bournville. May be you would have earned it.
You can also see Permutation and Combination – Fundamental Principle of Counting
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