Question

Find number of ways of selection of one or more letters from $AAAABBCCCDEF$
(i) there is no restriction.
(ii) the letters $A$ & $B$ are selected atleast once

Answer 1

There are $4$ $A$, $2B$, $3C$ $1D$, $1E$ and $1F$ then

$i)$ Total number of letters $=12$

Number of ways of selection of one or more letters ${=}^{12}{c}_1{+}^{12}{c}_2+......{+}^{12}{c}_{12}={\sum }_{r=1}^{12}{}^{12}{c}_r.$

$ii)$ There are $4$ ways of selecting $A.$

There are $2$ ways of selecting $B.$

There are $4$ ways of selecting $C$ as we select either $1C,or2Cor3Cor$ we don't choose $C$.

There are $2$ ways of selecting $D$ as we select either $1Dor$ we don't choose $D$.

There are $2$ ways of selecting $E$ as we select either $1Eor$ we don't choose $E$.

There are $2$ ways of selecting $F$ as we select either $1For$ we don't choose $F$.

Hence the total possible ways $=4\times 2\times 4\times 2\times 2\times 2=256.$