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

Step-by-step explanation:

GREENGAGE -->    GGG  R  EEE  N  A

The formula for a Permutation with duplicates is: $\dfrac{_nP_n}{a!\ ...\ z!}$

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:

$\dfrac{_9P_{9-2}}{3!\ 3!}\quad =\quad \dfrac{_9P_7}{3!\ 3!}\quad =\quad \large\boxed{5040}$