a Recall the formula for the number of permutations of n objects taken r at a time, where r ≤ n.
B
b Recall the formula for the number of permutations of n objects taken r at a time, where r ≤ n.
A
a 1320
B
b 95 040
Practice makes perfect
a We want to find the number of permutations of 3 letters taken from the Hawaiian alphabet, which has 12 letters. Let's recall the formula for the number of permutations of n objects taken r at a time, for 0 ≤ r ≤ n.
_n P_r = n!/(n-r)!
In our case, we choose from the 12 letters in the Hawaiian alphabet. Therefore, we can substitute 12 for n in the formula. We choose exactly 3 letters, so r = 3. Let's substitute these values into the formula.
We can evaluate the number of permutations _(12)P_3 using a graphing calculator. To do so, we have to start by entering the total number of letters in the Hawaiian alphabet, which is 12.
Next, we push MATH and then scroll right until we reach PRB. Then, scroll down to the second row and push ENTER.
Finally, by entering the number of letters to be chosen and hitting ENTER, we can calculate the desired number of permutations.
b This time we want to find the number of permutations of 5 letters taken from the 12 letters of the Hawaiian alphabet. We will use the same formula as in Part A.
_n P_r = n!/(n-r)!
In this case, we will substitute 5 for r. As in Part A, the value of n remains the same, 12.
There are 95 040 different permutations of 5 letters taken from the Hawaiian alphabet.
Alternative Solution
Using a Calculator
We can evaluate the number of permutations _(12)P_5 using a graphing calculator. To do so, we have to start by entering the total number of letters in the Hawaiian alphabet, 12.
Next, we push MATH and then scroll right until we reach PRB. Then, scroll down to the second row and push ENTER.
Finally, by entering the number of letters to be chosen and hitting ENTER, we can calculate the desired number of permutations.
