d What does the fact that some words are repeating mean for the number of combinations?
A
a 120 combinations
B
b 60 combinations
C
c 30 combinations
D
d The repeating words
Practice makes perfect
a There are 5 unique letters in the word ITEMS. Therefore, to determine how many arrangements there are we should calculate 5!. Start by entering 5 on your graphing calculator. Next, push MATH, scroll to the fourth menu, PRB, and choose the fourth option.
b Examining the word STEMS, we notice that one of the letters, S, is a repeat. Therefore, we cannot calculate the number of ways the letters can be arranged by a simple factorial. Instead we have to use the following formula.
n!/r_1!r_2!...
In this formula n is the number of items and r_1, r_2,... is the number of times that item 1, item 2, and so on, repeats. The word contains 5 letters, which means n= 5. Also, one word repeats twice, so r_1= 2.
5!/2!
Let's calculate this on a graphing calculator.
There are 60 combinations.
c Like in Part B, we have a word with five letters. However, in this word we have two letters that both repeat once, S EEM S. Therefore, we have to substitute n= 5, r_1= 2, and r_2= 2 in the formula from Part B.
5!/2! 2!
Let's calculate this on a graphing calculator.
There are 30 combinations.
d The difference lies in the repeating letters, which limits the number of distinct ways the letters can be arranged.
