Question

There are 20 books on Algebra and Calculus in our library. The greatest number of selections each of which consists of 5 books on each topic is possible only when there are _______ books on each topic in the library.

Answer 1

First we will first assume that there are $x$ algebra books and then use that to find the books available. Then we will compute the number of possible ways of selecting 5 from each using the combinations and then take $x$ such that ${}^x{C_5} \cdot {}^{20 - x}{C_5}$ is maximum when the value of $x$ is exactly half the value of 20 in the obtained expression to find the required value.

Complete step by step answer:

We are given that there are 20 books on Algebra and Calculus in the library.

Let us assume that there are $x$ algebra books.

Finding the number of calculus books, which are available, we get

$20 - x$

Computing the number of possible ways of selecting 5 from each using the combinations, we get

$\Rightarrow {}^x{C_5} \cdot {}^{20 - x}{C_5}$

We will take $x$ such that ${}^x{C_5} \cdot {}^{20 - x}{C_5}$ is maximum when the value of $x$ is exactly half the value of $20$, we get

$$\Rightarrow x = \dfrac{{20}}{2} \\\ \Rightarrow x = 10 \\\$$

Since ${}^{10}{C_5}$ is a maximum value of the above expression is maximum when $x$ is 10.

Hence, the final answer is 10.

Note: In solving these types of questions, students must know that while reading this problem one has to take care of all the steps to find the final answer. One should know the right time to use the formula of permutations or combinations, as a combination is a selection of an item from a collection, such that the order of selection does not matter, but a permutation is an arrangement of its member into a sequence or linear order. Students will try to solve the combination which won’t be necessary and will take a lot of time, as this question is a concept based not on calculations.