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  • When using the interior points of a triangle's three sides with AB, BC, and CA are 2, 4, and 6 as vertices, the total number of triangles that can be formed is equal to<div class=

When using the interior points of a triangle's three sides with AB, BC, and CA are 2, 4, and 6 as vertices, the total number of triangles that can be formed is equal to

Option: 1

144


Option: 2

169


Option: 3

196

 


Option: 4

154


Answers (1)

best_answer

A triangle can be created by selecting any one point from the three sides.

By selecting one point from one side and the remaining two points from either side, we can also create a triangle.

The number of possible ways to select 1 point out of 2 points from side AB is  ^{2}C_{1}

The number of possible ways to select 1 point out of 4 points from side BC is  ^{4}C_{1}

The number of possible ways to select 1 point out of 6 points from side CA is  ^{6}C_{1}

The number of possible ways to select 2 points out of 2 points from side AB is  ^{2}C_{2}

The number of possible ways to select 2 points out of 4 points from side BC is ^{4}C_{2}

The number of possible ways to select 2 points out of 6 points from side CA is ^{6}C_{2}

Thus, the total number of the triangle that will be formed is,

 

\begin{array}{r} { }^2 C_1 \times{ }^4 C_1 \times{ }^6 C_1+{ }^2 C_1 \times{ }^4 C_2+{ }^4 C_1 \times{ }^2 C_2+{ }^2 C_1 \times{ }^6 C_2+{ }^6 C_1 \times{ }^2 \ C_2+{ }^4 C_1 \times{ }^6 \\ C_2+{ }^6 C_1 \times{ }^4 C_2=2 \times 4 \times 6+2 \times 6+4 \times 1+2 \times 15+6 \times 1+4 \times 15+6 \times 6 \\ { }^2 C_1 \times{ }^4 C_1 \times{ }^6 C_1+{ }^2 C_1 \times{ }^4 C_2+{ }^4 C_1 \times{ }^2 C_2+{ }^2 C_1 \times{ }^6 C_2+{ }^6 C_1 \times{ }^2 C_2+{ }^4 C_1 \times{ }^6\\\\\ C_2+{ }^6 C_1 \times{ }^4 C_2=196 \end{array}

The total number of ways of constructing a triangle is 196 ways.

 

Posted by

Divya Prakash Singh

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