Mark the correct alternative in the following:

Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given point is

A. 216

B. 156

C. 172

D. none of these

Prove that the product of 2n consecutive negative integers is divisible by (2n)!

There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.[Hint: Required number = 210 – 1].

If ⁿ⁺¹C_r+1 : ⁿC_r = 11 : 6 and ⁿC_r : ^n–1C_r–1 = 6 : 3, find n and r.

If ⁿC_r–1 = 36, ⁿC_r = 84 and ⁿC_r+1 = 126, find r.

Fill in the Blanks

If ⁿP_r = 840, ⁿC_r = 35, then r = ______.

Fill in the Blanks

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 ______.

The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

The number of ways in which we can choose a committee from four men and six women so that the committee includes at least two men and exactly twice as many women as men is

The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is