There are $3$ books on mathematics $4$ on physics and $5$ on english .How many different collections can be made such that each collections consists of : Atleast one book of english

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Solution

All ways $= 2^{3} \times 2^{4} \times 2^{5}$
No of ways with at least one book of english $= 2^{3} \times 2^{4} \times 2^{5} - 2^{3} \times 2^{4}$
$= 2^{3} \times 2^{4} (32 - 1)$
$= 8 \times 16 \times 31 = 3968$

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There are 3 books on mathematics 4 on physics and 5 on english .How many different collections can be made such that each collections consists of : Atleast one book of english

All ways=23×24×25No of ways with at least one book of english =23×24×25−23×24=23×24(32−1)=8×16×31=3968

There are $3$ books on mathematics $4$ on physics and $5$ on english .How many different collections can be made such that each collections consists of : Atleast one book of english

All ways $= 2^{3} \times 2^{4} \times 2^{5}$
No of ways with at least one book of english $= 2^{3} \times 2^{4} \times 2^{5} - 2^{3} \times 2^{4}$
$= 2^{3} \times 2^{4} (32 - 1)$
$= 8 \times 16 \times 31 = 3968$