Question

Which among the following represents the selection of $5$ maths books from $20$ maths books and selection of $8$ english books from $15$ english books?

• A. ${}^{20}{C}_5\times {}^{15}{C}_8$
• B. ${}^{20}{C}_5+{}^{15}{C}_8$
• C. ${}^{20}{C}_{15}\times {}^{15}{C}_7$
• D. ${}^{20}{C}_{15}\times {}^{15}{C}_8$

Answer 1

Answer: (A), (C), (D)

${}^{20}{C}_{15}\times {}^{15}{C}_8$

Total number of math books $=20$
We want to select $5$ math books from $20$ math books. This can be done in ${}^{20}{C}_5$ ways.

We know that
${}^n{C}_r={}^n{C}_{n-r}$

$⟹{}^{20}{C}_5={}^{20}{C}_{20-5}={}^{20}{C}_{15}$


Total number of english books $=15$
We want to select $8$ english books from $15$ english books. This can be done in ${}^{15}{C}_8$ ways.

We know that
${}^n{C}_r={}^n{C}_{n-r}$

$⟹{}^{15}{C}_8={}^{20}{C}_{15-8}={}^{15}{C}_7$

The selection of $5$ math books from $20$ math books and selection of $8$ english books from $15$ english books would be obtained by multiplying ${}^{20}{C}_5$ by ${}^{15}{C}_8.$

Number of ways of selecting $5$ math books from $20$ math books and selecting $8$ english books from $15$ english books:
${}^{20}{C}_5\times {}^{15}{C}_8$

But,
${}^{20}{C}_5\times {}^{15}{C}_8={}^{20}{C}_5\times {}^{15}{C}_7$

$={}^{20}{C}_{15}\times {}^{15}{C}_7={}^{20}{C}_{15}\times {}^{15}{C}_8$

Therefore, option (a.), (c.) and (d.) are the correct answers.