Question

How many ways can the letters in the word GAME be arranged if the letters A and M have to be together?

a) 6 ways
b) 12 ways
c) 24 ways
d) 48 ways

Answer 1

There are 12 ways to arrange the letters in the word GAME with A and M together. This is derived by treating the pair AM as a single unit, arranging the 3 units, and accounting for the interchangeability of A and M.

Step-by-step explanation:

To find the number of ways to arrange the letters in the word GAME with A and M together, one can treat the pair AM or MA as a single unit. Thus, instead of arranging 4 letters, we arrange 3 units: AM/GM, E. Since there are 3 units, there are 3! (3 factorial) ways to arrange them, which is 3 x 2 x 1 = 6 ways. However, since A and M can also be switched around within their unit, we must multiply this number by 2. Therefore, we have a total of 6 x 2 = 12 arrangements.

Here's the step-by-step logic:

1. Treat AM as one unit.
2. Arrange the units: 3! = 6 ways.
3. Consider that AM can also be written as MA: multiply by 2.
4. The final calculation: 3! x 2 = 12 ways.