10 choices how many combinations

Permutations with repetitions How many ways are there to choose 4 balls out of our 5 balls from before if we can repeat ourselves? Well in our first selection we would still have 5 options. In our second selection, we would again have five option because we have replaced our last choice, so any of the 5 balls can come next.

The same would apply to the third and fourth selections. If we now had to choose 3 balls instead, it would be 5 3 permutations. In general, if we have n balls and we want to choose r balls with repetitions we have n r different permutations. Combinations - selections when order is not important 3. Combinations without repetitions How many ways are there to choose 3 balls out of our 5 balls from before if we can't repeat ourselves and we don't care about the order?

We can start off by seeing that the problem is similar to counting permutations without repetitions. Let's take a specific combination, say the balls 1, 2 and 3. How many times does the combination appear in our permutation list? That's because there are 3! So if we choose 3 balls out of 5 with no repetitions, we have 5! Combinations with repetitions How many ways are there to choose 8 balls out of the 5 balls if we can repeat ourselves?

We divide by r! A group of 12 women and 5 men are used to pick a committee of 6 people. What is the possible outcomes if. Since order does not matter and there is no replacement, we use combinations.

Class exercise: Find n C r , for. Exponentials and Logarithms: inverses of each other, irrational number e, using calculators, rewriting each in terms of the other. Properties of Logarithms: Inverse properties to solve equations, 3 rules of logs. Exponential Growth: Average growth rate, exponential growth model, population, appreciation in real estate. Exponential Decay: Exponential Decay model, carbon-dating, half-lifes, radioactive decay.

Tree diagrams, shortcuts, Factorials. Permutations and Combinations: n P r , n C r. Choosing without replacement, order matters, order does not matter. Back to Counting and Probability Main Page. Example 1: A PIN code at your bank is made up of 4 digits, with replacement. Example 2: There are 10 entries in a contest. When order of choice is not considered, the formula for combinations is used. Now suppose that you were not concerned with the way the pieces of candy were chosen but only in the final choices.

In other words, how many different combinations of two pieces could you end up with? In counting combinations, choosing red and then yellow is the same as choosing yellow and then red because in both cases you end up with one red piece and one yellow piece. Unlike permutations, order does not count. Table 3 is based on Table 2 but is modified so that repeated combinations are given an "x" instead of a number. For example, "yellow then red" has an "x" because the combination of red and yellow was already included as choice number 1.

As you can see, there are six combinations of the three colors. As an example application, suppose there were six kinds of toppings that one could order for a pizza. How many combinations of exactly 3 toppings could be ordered? The formula is then:. Table 1. Six Possible Orders.

Number First Second Third 1 red yellow green 2 red green yellow 3 yellow red green 4 yellow green red 5 green red yellow 6 green yellow red. Table 2. Twelve Possible Orders.