Question

There are $3$ sections in a question paper each containing $5$ questions. A candidates has to solve only $5$ questions, choosing at least one question from each section. In how many ways can he make his choice?

• A. ${}^{15}{C}_5$
• B. ${}^3{C}_1\times {}^{12}{C}_4$
• C. $2250$
• D. None of these

Answer 1

Answer: (C)

$2250$

Let there are 3 section namely a,b,c. Each having 5 question. A candidate has to solve only 5 question. Candidate can choose 'one' que. from section 'a', 'one' question from section 'b' & 'three' question from section 'c' Which can be choosen in$-5c^1\ast 5c^1\ast 5c^3$ ways. Again, 'one' que from section 'a', 'three' ques from section 'b', & 'one' ques section 'c'. Which can be choosen in $-5c^1\ast 5c^3\ast 5c^1$ ways Again, 'three'question from section 'a', 'one'que from section 'b', & 'one' que from section 'c' Which can be choosen in$-5c^3\ast 5c^1\ast 5c^1$ Candidate can also choose 'two' que from section 'a', 'two' que from section'b', & 'one' que from section 'c' Which can be choosen in$-5c^2\ast 5c^2\ast 5c^1$ ways Again, 'two'que from section 'a', 'two' que from section'b', & 'one'que from section 'c'. Which can be choosen in$-5c^2\ast 5c^2\ast 5c^1$ ways Again, 'one'que from section 'a', 'two' que from section 'b',& 'two' que from section'c' Which can be choosen in- $5c^1\ast 5c^2\ast 5c^2$ Total no. Of ways to choose 5 questions - $(5c^1\ast 5c^1\ast 5c^3)\ast 3$ + $(5c^1\ast 5c^2\ast 5c^2)$ = $2250$