Question:

There are three coplanar parallel lines. If any $P$ points are taken on each of the lines, the maximum number of triangles with vertices at these points, is

Updated On: Jul 7, 2022

$^{3P}C_{3}$
$P^{2} \left(P-1\right)$
$P^{2} \left(4P-1\right)$
$P^{2} \left(4P-3\right)$

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The Correct Option is D

Solution and Explanation

Required number of ways $=\, ^{P}C_{1} \times\, ^{P}C_{1} \times\,^{P}C_{1} + \,^{P}C_{2} \times\,^{2P}C_{1} \times\, ^{3}P_{1}$ $ = P^{2}\left(4P-3\right)$.

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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence.

In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.

Combination refers to the combination of about n things taken k at a time without any repetition.
The combination is used for a group of data where the order of data does not matter.