State whether the statements in True or False.

There are 12 points in a plane of which 5 points are collinear, then the number of lines obtained by joining these points in pairs is ¹²C₂ – ⁵C₂.

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A committee of 6 is to be chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee.

[Hint: At least 3 men and 2 women: The number of ways = ¹⁰C₃ × ⁷C₃ + ¹⁰C₄ × ⁷C₂.

For 2 particular women to be always there: the number of ways = ¹⁰C₄ + ¹⁰C₃ × ⁵C₁.

The total number of committees when two particular women are never together = Total – together.]

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A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ______.

State whether the statements in True or False.

Three letters can be posted in five letterboxes in 3⁵ ways.

State whether the statements in True or False.

In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is ^n–mP_r–m × ^rP_m.