Question

There are 20 books on Algebra and Calculus in one library. For the greatest number of selections each of which consists of 5 books on each topic. The possible number of algebra books is N then the value of $\dfrac{N}{2}$ is.

Answer 1

As the total number of books is 20, we can take the number of books on Algebra as x, and the number of Calculus books will be 20-x. Now we check the combination of taking 5 books of each and find the value of x when the number of combinations is maximum. Half of the x value we obtained will be the required answer.

Complete step by step solution: As per the question, we know that there are a total of 20 books. So, if the number of books on Algebra is taken as x, then the number of algebra books will be 20-x.
We need to take 5 books for each subject. Its possible combinations are given by,
${}^{\text{x}}{{\text{C}}_{\text{5}}} \times {}^{{\text{20 - x}}}{{\text{C}}_{\text{5}}}$
The value of the above combination is maximum when x = 10 because ${}^{{\text{10}}}{{\text{C}}_{\text{5}}} \times {}^{{\text{10}}}{{\text{C}}_{\text{5}}}$ has the maximum value for the combination.
So, the number of books in Algebra, ${\text{N = 10}}$.
So, $\dfrac{{\text{N}}}{{\text{2}}}{\text{ = 5}}$

Therefore, the required solution is 5.

Note: We have used the concept of permutations and combinations to solve the problem. Here we took the combination of selecting 5 books of each subject. We needed the value of the combination to be maximum. The permutation is used when we need to do arrangements among some items. Whereas, the combination is used to choose or pick something among some items.