a Is the order in which we choose students important? Should permutations or combinations be used?
B
b Is the order in which we choose students important? Should permutations or combinations be used?
A
a 4060
B
b 142 506
Practice makes perfect
a We want to find the number of 3-student committees we can form out of a class of 30 students. Note that the order we choose the students in is not important since 3 students are in the committee, no matter the order they were chosen in. Since the order is not important, let's use combinations.
_nC_r = n!/r! (n-r)!
The formula for the number of combinations of n objects taken r at a time, for 0 ≤ r ≤ n. In our case, we are choosing from the class of 30 students, so n = 30. Out of these, we choose exactly 3 students to form the committee, so r = 3. Let's substitute these values into the formula.
The committee can be formed in 4060 different ways.
Alternative Solution
Using a Calculator
We can evaluate the number of combinations _(30)C_3 using a graphing calculator. To do so, we have to start by entering the total number of students, 30.
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 students to be chosen and hitting ENTER, we can calculate the desired number of combinations.
b This time we want to find the number of different ways we can form a committee of 5 students. We will use the same formula as in Part A.
_nC_r = n!/r! (n-r)!
In this case, we will substitute 5 for r. The value of n remains the same as in Part A, 30.
There are 142 506 different ways the committee of 5 students can be formed.
Alternative Solution
Using a Calculator
We can evaluate the number of combinations _(30)C_5 using a graphing calculator. To do so, we have to start by entering the total number of students, 30.
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 students to be chosen and hitting ENTER, we can calculate the desired number of combinations.
