Question

There are 18 books on a shelf: 6 mathematics, 4 physics, 5 history, and 3 biology. If Emil picks six books at random, what is the probability that he picks 3 mathematics,  2 physics and one other book?

Answer 1

Since order of selection doesn't matter hence we will use combination instead of permutation.

As per multiplication theorem rule,

If a work w1 can be done in n1 number of ways and a work w2 can be done in n2 number of ways .

Then, number of ways to perform both works w1 and w2 = n1*n2

Also, out of n objects, r (r< n) objects can be selected in nCr ways.

Similarly, here 3 mathematics books can be selected in 6C3 = 20 ways

2 physics books can be selected in 4C2 = 6 ways

1 other book can be either a history or a biology book

That is one other book can be selected in (5+3)C1 = 8 ways.

So, number of ways of selecting 3 mathematics books, 2 physics books and 1 other book = 20*6*8 = 960 ways

and total number of ways of selecting 6(= 3+2+1) books out of 18(=6+4+5+3) books = ​​​​​18C6 = 18564 ways

Hence, probabillity of selecting 3 mathematics books, 2 physics books and 1 other book = 960/18564 = 80/1547 = 0.051713

So, answer is 0.051713 or 5.17%

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