Is the order in which we choose the singers for the quartet important? Should permutations or combinations be used?
1/1365
Practice makes perfect
We want to find the probability of choosing the youngest four members from a 15-member choir for a quartet. To do so, we will use the theoretical probability formula.
P=Favorable Outcomes/Possible Outcomes
We start by finding the number of possible outcomes. We want to find how many ways we can choose a quartet from a 15-member choir. Notice that the order in which we choose the singers is not important because the 4 selected singers are in the quartet, regardless of the order in which they were chosen.
_nC_r = n!/r! (n-r)!
In the formula for the number of combinations, n is the total number of objects taken r at a time, for 0 ≤ r ≤ n. In our case, we are choosing from a 15-member choir, so n = 15. Of these, we are choosing exactly 4 singers to form the quartet, so r = 4. Let's substitute these values into the formula.
The total number of possible outcomes is 1365. Next, we will look for the number of favorable outcomes. Note that there is only one combination in which all four youngest members of the choir will be selected for the quartet, so the number of favorable outcomes is equal to 1. We have enough information to calculate the desired probability.
P=Favorable Outcomes/Possible Outcomes [0.9em]
⇕
P=1/1365
Alternative Solution
Using a Calculator
We can evaluate the number of combinations _(15)C_4 using a graphing calculator. To do so, we have to start by entering 15.
Next, we push MATH and then scroll right until we reach PRB. Then, scroll down to the third row and push ENTER.
Finally, by entering the number of singers to be chosen and hitting ENTER, we can calculate the desired number of combinations.
