Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged under given conditions? 1. if exactly two of the Gs are next to each other. **Question**

$(1)$ From the word $GREENGAGE$; lets arrange the word this way $AEEEGGGNR$.
$(2)$ Now let's ignore the $G$'s for now. You now have: $AEEENR$.
$(3)$ Now add the $G$'s that are stuck together. You get: $AEEE(GG)NR$.
$(4)$ So now you can take that as 7 letters so we can arrange them in $7!$ ways. We also have to divide it $3!$ Because of the $3E$'s.
$(5)$ Now between these letters try and insert the extra $G$ between letters except next to the (GG). There are $6$ possible ways.

> Hence, we have a total of $\frac{7!}{3!}\times 6=5040$ ways.

Hope it helps.