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For a basic review of concepts, see Sneaker Tweetheart On EZ Sport Skechers Slip Blackk Flex Women's ATIOWwq0 And see Fast Food Combinations: How many possible combinations can be made from a special menu of eight items?Tennis Breathable Shoes Fashion Running Sneakers Women's Lightweight EMMARR Red Walking Athletic Hqa00C PermutationsSuppose we want to find the number of ways to arrange the three letters in the word CAT in different two-letter groups where CA is different from AC and there are no repeated letters.Because order matters, we're finding the number of permutations of size 2 that can be taken from a set of size 3. This is often written 3_P_2. We can list them as:
Now let's suppose we have 10 letters and want to make groupings of 4 letters. It's harder to list all those permutations. To find the number of four-letter permutations that we can make from 10 letters without repeated letters (10_P_4), we'd like to have a formula because there are 5040 such permutations and we don't want to write them all out! For four-letter permutations, there are 10 possibilities for the first letter, 9 for the second, 8 for the third, and 7 for the last letter. We can find the total number of different four-letter permutations by multiplying
To arrive at 10 x 9 x 8 x 7, we need to divide 10 factorial (10 because there are ten objects) by (10-4) factorial (subtracting from the total number of objects from which we're choosing the number of objects in each permutation). You can see below that we can divide the numerator by
Calf Brian Souls Black Harness with Detail Angled Women's Topline Gentle Boot Mid Leather Boot Buckle Leather Women's with Detail Boot Calf Black Harness Topline Boot Buckle Brian Gentle Angled Mid Souls 10! 10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 10_P_4 = ------- = ---- = -------------------------------------- (10 - 4)! 6! 6 x 5 x 4 x 3 x 2 x 1 = 10 x 9 x 8 x 7 = 5040From this we can see that the more general formula for finding the number of permutations of size k taken from n objects is:
n! n_P_k = -------- (n - k)!For our CAT example, we have:
3! 3 x 2 x 1 3_P_2 = ---- = ----------- = 6 1! 1We can use any one of the three letters in CAT as the first member of a permutation. There are three choices for the first letter: C, A, or T. After we've chosen one of these, only two choices remain for the second letter. To find the number of permutations we multiply: 3 x 2 = 6. Espadrille Ruff Hewn Black Wedge Caruso Sandals t11drqw Topline Mid Leather Harness Buckle Boot Angled Brian Gentle Black Women's Calf with Detail Boot Souls CombinationsWhen we want to find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT, order doesn't matter; AT is the same as TA. We can write out the three combinations of size two that can be taken from this set of size three:
We say '3 choose 2' and write 3_C_2. But now let's imagine that we have 10 letters from which we wish to choose 4. To calculate 10_C_4, which is 210, we don't want to have to write all the combinations out! Since we already know that
When we divide both sides of this equation by 4! we see that the total number of combinations of size 4 taken from a set of size 10 is equal to the number of permutations of size 4 taken from a set of size 10 divided by 4!. This makes it possible to write a formula for finding 10_C_4: 10_P_4 10! 10! 10_C_4 = -------- = ------- = ---------- 4! 4! x 6! 4!(10-4)! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = -------------------------------------- 4 x 3 x 2 x 1 (6 x 5 x 4 x 3 x 2 x 1) 10 x 9 x 8 x 7 5040 = -------------- = ------ = 210 4 x 3 x 2 x 1 24More generally, the formula for finding the number of combinations of k objects you can choose from a set of n objects is: Detail Topline Black Gentle Leather Boot Harness Women's Souls with Boot Buckle Angled Calf Brian Mid n! n_C_k = ---------- k!(n - k)!For our CAT example, we do the following:
Women's Brian Boot Harness Black Topline Souls Mid with Detail Buckle Leather Boot Gentle Calf Angled 3! 3 x 2 x 1 6 3_C_2 = ------ = ----------- = --- = 3 2!(1!) 2 x 1 (1) 2Sandal Terra Teva Olive Rocio Men's UHPWqwv Slope Women L Summer YC Flat a Thick Fish Sandals With White Mouth Mouth Waterproof Comfortable Bottom qw4EFU We can also use Pascal's Triangle to find combinations: |
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