Question

Find the number of ways in which one or more letters can be selected from the letters
$AAAAABBBBCCCDDEFG$.

Answer 1

Here $5A^{\prime }s$ are alike, $4B^{\prime }s$ are alike, $3C^{\prime }s$ are alike, $2D^{\prime }s$ are alike and $E,F,G$ are different.
Then the total number of combinations
$=(5+1)(4+1)(3+1)(2+1)2^3-1=\mathrm{6.5.4.3.2.8}-1=2879$