Welcome to combinations! A combination is a way of selecting items from a collection where the order doesn't matter. For example, if we're choosing 3 people for a committee from 5 candidates, selecting Alice, Bob, and Carol is the same as selecting Bob, Carol, and Alice. The order of selection doesn't change the committee itself.
The combination formula is C of n comma k equals n factorial divided by k factorial times n minus k factorial. Here n is the total number of items and k is the number we want to choose. We use factorials because we start with permutations which count order, then divide by k factorial to remove the ordering arrangements. Let's see an example: C of 5 comma 3 equals 5 factorial divided by 3 factorial times 2 factorial, which equals 120 divided by 6 times 2, giving us 10 different combinations.
Let's work through a complete example step by step. We want to choose 3 fruits from 5 available fruits for a fruit salad. First, we identify n equals 5 total fruits and k equals 3 fruits to choose. Next, we apply the combination formula: C of 5 comma 3 equals 5 factorial divided by 3 factorial times 2 factorial. We calculate the factorials: 5 factorial is 120, 3 factorial is 6, and 2 factorial is 2. Finally, we get 120 divided by 12, which equals 10 different combinations. Here you can see all 10 possible fruit combinations displayed.
Now let's understand the key difference between combinations and permutations. In combinations, order does not matter - we're simply selecting items. The formula is n factorial divided by k factorial times n minus k factorial. For example, choosing 3 people for a committee. In permutations, order does matter - we're arranging items in specific positions. The formula is n factorial divided by n minus k factorial. For example, arranging 3 people in specific seats. The relationship between them is that permutations equal combinations times k factorial, because each combination can be arranged in k factorial different ways.
Combinations have many real-world applications. In lotteries, we calculate the odds by finding how many ways we can choose 6 numbers from 49, which gives us almost 14 million combinations. In sports, coaches use combinations to select teams - choosing 5 players from 12 gives 792 possible teams. Restaurants use combinations for menu options - 3 toppings from 8 choices creates 56 different combinations. Remember: combinations are for selection problems where order doesn't matter, they use the formula n factorial divided by k factorial times n minus k factorial, and they're always smaller than permutations. Combinations help us solve many counting problems in everyday life!