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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.

Respuesta :

tramserran

Answer:  5040

Step-by-step explanation:

GREENGAGE -->    GGG  R  EEE  N  A

The formula for a Permutation with duplicates is: [tex]\dfrac{_nP_n}{a!\ ...\ z!}[/tex]

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: [tex]_nP_{n-2}[/tex]

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

                                      and 3! to represent the E's

Solution:

[tex]\dfrac{_9P_{9-2}}{3!\ 3!}\quad =\quad \dfrac{_9P_7}{3!\ 3!}\quad =\quad \large\boxed{5040}[/tex]

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