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Calculating Combinations. Suppose we have determined that we need to use a combination in a particular question. Sometimes the answer choices will be written in the form, say, a 50 choose 10. That will actually be one of the answer choices. So, we won’t actually have to calculate it.
That most typically happens when the combinations are very large and hard to calculate. The test is not interested in making us do really lengthy calculations. Much more interested, simply, can we recognize the proper answer and then they give it to us in, in the form of n choose r, something like 50 choose 10. More often though, we would need to calculate the combinations.
First of all, remember that the formula it desi, designed to begin with a massive amount of calculation. If you ever make the mistake of calculating the three factorials separately, you will be making the problem a hundred times harder for yourself than it has to be. Using this formula is actually not the most efficient way to approach most combination problems, and so that’s why I have not been emphasizing that formula at all.
The approach we have recommended in a previous video is much more efficient. Begin with the fundamental counting principle, then divide to eliminate the repetitions. Remember to cancel before you multiply. Remember to leave all products in unmultiplied form until all possible canceling is done.
Very important. Now, in addition to that, I will also show a couple other ways to calculate combinations. One shortcut involves the case of n choose 2, that is, the case in which we are picking an unordered pair from a larger set. Let’s think about this with the fundamental counting principle approach.
We have n choices for the first item and n minus 1 for the second. But because the order of selection doesn’t matter, we have to divide by 2 factorial, which is just 2. So n choose 2 would be n times n minus 1, divided by 2. You may remember from the sum of sequences, that this is also the sum of the first n minus 1 positive integers.
It’s a popular little formula. It’s worth remembering that this shortcut pattern shows up in a couple different areas of mathematics, in kind of keeping everything straight. Another mathematical prat, pattern that is directly related to n choose r, is Pascal’s triangle, named for Blaise Pascal. So, this may be familiar.
This is actually something they often show students in grade school because it’s easy practice with addition. The two outside diagonals are all 1s, and each number on the inside is the sum of the two diag, the two numbers diagonally above it. So, for example, the 4 here, is the sum of the 1 and the 3. The 10 is the sum of the 6 and the 4.
The 15 is the sum of the 10 and the 5, so you keep on creating sums all the way down. Pascal triangle gives all the values of n choose r. In fact, n choose r is the entry in the nth row, at the rth place in Pascal’s triangle. The tricky thing about this is that we need to remember to start counting rows and places from zero.
The top one is the 0th row, the 0th place, 0 choose 0 equals 1, but that’s not very meaningful in real-world combinations. In any row, the second number in the row gives you the row number. Here is the 7th row of Pascal’s triangle. The first entry is 7 choose 0, which equals 1, not very interesting in real-world combination, but the others are interesting.
7 choose 1, of course, is 7. Then 7 choose 2 is 21, 7 choose 3 and 7 choose 4 are both 35, 7 choose 5 is 21, 7 choose 6 is 7, and then 7 choose 7 has to be 1. There’s only one way to pick all seven people from a group of seven. If you get quick at, if you practice constructing Pascal’s triangle, you will get quick at it.
In particular, it might helpful, say, to memorize the fourth or the fifth row. If you memorize that row, then you could just jot that down and do all, quickly do the rows immediately below it. And if you do this at the beginning at the quant section and just have it for reference in case combinations come up, that can be one quick way to think about these numbers.
If, in a problem, you have to figure out different-size combinations from the same pool, say, some situation in which you have to choose, you have to calculate both 8 choose 3 and 8 choose 4, it may be quicker to draw a Pascal’s triangle down to that row rather than do the separate calculations. Remember that you can always calculate combinations by using the fundamental counting principle and dividing to eliminate the repetitions.
That’s what we’ve shown in previous videos. Remember to cancel if you use the nCr formula, the combination formula, including canceling before you multiply. Remember the shortcut formula for n choose 2, and practice using Pascal’s triangle to generate values of n choose r.
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