Priya and Padmini solve a problem on “How to count without actually counting” given in the text book as follows:
Find the number of ways in which a selection of 5 books can be done out of 3 physics, 3 chemistry and 3 mathematics books of taking at least one book of each subject, books of same subject being different.
Priya’s Solutions:
One book of each subject can be taken in $^{3} C_{1}$ ways balance 2 books from remaining 6 books can be taken in $^{6} C_{2}$ ways hence total no.of ways using fundamental principle of counting . $=^{3} C_{1} .^{3} C_{1} .^{3} C_{1} .^{6} C_{2}$
Padmini’s Solutions:
Number of ways in which atleast one Physics book can be selected $= (2^{3} - 1)$ perpendicularly one chemistry in $= (2^{3} - 1)$ ways and one mathematics in $= (2^{3} - 1)$ ways.
Total ways $= (2^{3} - 1) (2^{3} - 1) (2^{3} - 1)$
Which one of the following options is correct?

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Priya is correct and Padmini is wrong.

Padmini is correct and Priya is wrong.

Both Priya and Padmini are wrong and their answers are higher than the actual answer.

Both Priya and Padmini are wrong and their answers are lower than the actual answer.

answer is C.

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Detailed Solution

Number of ways $=^{9} C_{5}$ - no.of ways in which she selects 5 from only of two categories.
$=^{9} C_{5} - 3.^{6} C_{5} = 126 - 18 = 108$
Alternatively: Number of ways 1 1 3 or 2 2 1 i.e., .
$3.^{3} C_{1} .^{3} C_{1} .^{3} C_{3} + 3.^{3} C_{2} .^{3} C_{2} .^{3} C_{1} = 27 + 81 = 108$