Suppose there are $6$ books on Maths, $3$ books on English and $2$ books on science . How many ways to select atleast one book of each subject, assuming the books of same subject are different ?
My try:
I have not solved it, but I just need to check my logic.
Number of ways:-
Total ways to select $3$ books – (Total ways to select $3$ books on Maths + Total ways to select $3$ books on English + Total ways to select $3$ books on Science + Total ways to select $3$ books on Maths and English + Total ways to select $3$ books on Maths and science + Total ways to select $3$ books on Science and English)
Is this logic right ?
Best Answer
Select one or more Maths books. Number of ways:
$\dbinom{6}{1}+\cdots+\dbinom{6}{6}=2^6-1$
Select one or more English books. Number of ways:
$\dbinom{3}{1}+\cdots+\dbinom{3}{3}=2^3-1$
Select one or more Science books. Number of ways:
$\dbinom{2}{1}+\cdots\dbinom{2}{6}=2^2-1$
So, required ways
$(2^6-1)(2^3-1)(2^2-1)$