From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and

[BTGmoderatorDC]

From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720


OA A

Source: Magoosh

[Scott@TargetTestPrep]

From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720


OA A

Solution:

We see that MAGOOSH has 7 letters in total and 6 distinct letters.

If the three-letter word to be formed consists only of distinct letters, then there are 6P3 = 6 x 5 x 4 = 120 such words.

If the three-letter word to be formed consists of letters that are not all distinct, then the two Os must be in it. If the two O’s along with one of the other 5 letters are in the word, then the number of words can be formed is 5 x 3!/2! = 5 x 3 = 15.

Therefore, the total number of words that can be formed is 120 + 15 = 135.

Alternate Solution:

The number of words that do not include the letter O is 5P3 = 5!/2! = 5 x 4 x 3 = 60.

The number of words that include one O is 5C2 x 3! = [(5 x 4)/2] x 6 = 60.

The number of words that include two Os is 5C1 x 3!/2! = 5 x 3 = 15.

We see that there are a total of 60 + 60 + 15 = 135 three letter words.

Answer: A

[swerve]

From the letters in MAGOOSH, we are going to make three-letter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different three-letter words can be made from the seven letters in MAGOOSH?

A. 135
B. 170
C. 123
D. 121
E. 720


OA A

Source: Magoosh

First, take all different letters: M A G O S H, so 6 letters.
They can be arranged in \(6\cdot 5\cdot 4=120\) ways

Second, two are O and third any of remaining 5.
So \(5\cdot \dfrac{3!}{2!}=15\) as each combination can be arranged in \(\dfrac{3!}{2!}\) ways

Total \(120+15=135\)

Therefore, A