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A systematic procedure for finding all combinations of n objects taken r at a time

The following is a systematic procedure for finding and listing all combinations of n objects taken r at a time:

Step 1. Number the objects — give them numbers from 1 to n. This reduces the problem to that of finding all combinations of the integers 1, 2 ... , n, taken r at a time.

Step 2. Compute the number of combinations there will be from the formula

Step 3. Start with the combination 123...r. From there start listing easy and obvious combinations as in the following example.

Example. Find all combinations of integers 1, 2, 3, 4, 5 taken 3 at a time. The number of
combinations will be _{n}C_{r} = _{5}C_{3 }= 5!/2!3! = 10. Starting with the combination 123 we write:

123

124

125

134

135

145

Now that we have six easy combinations written down we utilize the following theorem to find the remaining four combinations.

Theorem 1. In the final list of combinations all integers will occur the same number of times.

In the above example the full list of combinations happens to be

123

124

125

134

135

145

234

235

245

345

Let us count the number of times the number 1 occurs in the list, the number of times the number 2 occurs in the list, the number of times the number 3 occurs in the list, etc. If we do this we will discover that the number 1 occurs 6 times, the number 2 occurs 6 times, etc. Each of the numbers 1 - 5 occurs six times in the list. We use this fact in trying to figure out what the remaining combinations are. Returning now to our example we have the following combinations

123

124

125

134

135

145

and want to figure out what the others are. We note that 1 occurs six times, 2 occurs three times, 3 occurs three times, 4 occurs three times and 5 occurs three times. If 1 occurs 6 times, then all the others must occur at least 6 times in the final list. Let us now start writing some numbers starting with the digits 23. We write the numbers 234 and 235 to give

123

124

125

134

135

145

234

235

We count digits again and find now that 2 occurs 5 times, 3 occurs 5 times, 4 occurs 4 times, and 5 occurs 4 times. We have eight combinations with only two combinations remaining and this information gives us a good idea of what they are. Let us try the number 245. It is a valid combination that doesn’t appear elsewhere in the list. Now the only digits left are 3, 4, and 5 — which give us the last number. Thus we get the full list

123

124

125

134

135

145

234

235

245

345

When we write down the combinations we follow the practice of writing them in order of increasing magnitude of the integers i.e. we write a combination as 345 instead of as 543.

Observation of the pattern that the numbers follow in the above list can be helpful in helping us guess at combinations.

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