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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 ?

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    what is the answer you get in this appraoch? I get $1323$ in the way i explained. – 2017-01-22
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    also, question says at least one book of each subject must be there. so, 3 or more books can be selected. your approach does not count selection of more than 3 books. – 2017-01-22

2 Answers 2

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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
$(2^6-1)(2^3-1)(2^2-1)$

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    I think this method not works here. If user require maximum 5 books are selected and have condition for particular books. – 2017-01-22
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    @KanwaljitSingh, pl see the edit – 2017-01-22
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I think its not work. As you are not including one subject in all 3 cases.

I think more easy way is to find -

Total cases - (Cases with only 1 subject books selected + 2 subjects books selected)

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    I h'v edited !! Can you check now ? – 2017-01-22
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    Yes its looking perfect now. – 2017-01-22
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    @KanwaljitSingh, question says at least one book of each subject. so, selection 4 or more books can also be valid , isn't it. so how do you find the total cases? also there are other cases you need to subtract, like 'selection of 3 books where all books are English' – 2017-01-22
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    Sorry typing mistake. – 2017-01-22