Combinatorial problems are difficult because it’s hard to know which formula to use in a particular problem and when you need to ‘tweak’ or totally abandon the formula. In this post I share two solutions to a problem which connects the multiplication principle, the combination formula and the formula for counting the number of permutation with repetition. Knowledge of connections among concepts help in problem solving. The problem is generated from the rook puzzle presented in my post Connecting Pascal’s Triangle and Permutation with identical objects.
Find the number of different ways of arranging 14 letters 7 of which are E’s and 7 are N’s, in a row. Here is one arrangement:
The first solution shows how the formula for counting permutation with identical objects can be deduced from the solution involving the multiplication principle and the second connects permutation and combination.
The idea behind this solution is to initially treat each letter as distinct. There are 14 letters to be arranged in a row. If these letters are distinct from one another then by the Multiplication Principle, there are 14! different arrangements in all.
But the letters are not all distinct. In fact there are only two kinds – N’s and E’s. This means an arrangement, for example, consisting of 7 N’s followed by 7 E’s,
N E N E N E N E N E N E N E
has been counted 7!7! times in all in 14!. The same is true for this arrangement:
N N N E E N N E E N N E E E.
Thus, to find the number of ways of arranging 14 letters where 7 of which are identical and the remaining 7 are also identical, 14! need to be divided by 7!7!
If the problem had been In how many ways can you arrange 10 N‘s and 4 E‘s?, the solution will be .
Notice that this solution uses the technique of counting the number of permutations (arrangements) of n objects, r1 of which are identical, r2 are identical, . . . , and rn are identical, where r1 + r2 + . . . ri = n. The number of different permutations is denoted by
In the problem, n = 14, r1 = 7 and r2 = 7. Hence, the total number of arrangements is .
This solution simplifies the original problem to How many different ways can 7 N’s be arranged in a row of 14 spaces? Now, why is this problem equivalent to the original problem? What happened to the 7 E‘s? Why aren’t they not considered anymore? This is because for a particular arrangement of 7 N‘s in 14 spaces, there is one and only one way the 7E‘s can be arranged.
To solve, count the number of possible positions for the 7 N’s. You have 14 positions to choose from for the first N. For the next N you only have 13 positions to choose from, for the next N, 12 and so on until the 7th N. By the Multiplication Principle you have 220.127.116.11.10.9.8.7 different possible positions where you consider each N to be distinct.
But the N’s are identical. That is, in the arrangement for example
_ N _ N N N _ N N _ _ _ _ N
N has been counted 7! times in 18.104.22.168.10.9.8.7. So you have to divide this by 7!
If there were 10N‘s and 4 E‘s, the problem would have been In how many ways can I arrange 10 N’s in 14 spaces?
In general, if there were n possible positions for arranging r objects, the formula is . Note that this looks like the combination formula which is used to solve problems for the number of combination and indeed it is. You just got used to applying to problems like In how many way can you arrange n different objects taken r objects at a time?