Question

Q: Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged if exactly two of the Gs are next to each other.

Its answer is 5040, but I don't get the method.

Answer 1

Answer: 5040

Explanation:

GREENGAGE --> GGG R EEE N A

The formula for a Permutation with duplicates is:
`(_nP_n)/(a!\ ...\ z!)`

Where

• n: the total number of terms
• a: the number of duplicates for one of the terms
• z: is the number of duplicates for a different term

Since we have a requirement that 2 of the G's must be next to each other, the numerator reflects as:
`_nP_(n-2)`

The denominator will have 3! to represent the G's

and 3! to represent the E's

Solution:

`(_9P_(9-2))/(3!\ 3!)\quad =\quad (_9P_7)/(3!\ 3!)\quad =\quad \large\boxed{5040}`