CRE 10 - Geometry based questions | Modern Math - Permutation & Combination

Answer: 96

Explanation :

If we select any two non-collinear points we will get a different line except when we select the 5 collinear points, we will get the same line again and again.

Total number of ways of choosing 2 points = ¹⁵C₂ = 105.

For collinear set of points, number of selecting 2 points = ⁵C₂ = 10. 

Now each of these 10 lines is same hence, we will subtract these 10 lines and add 1 line.

∴ Required answer = 105 – 10 + 1 = 96.

Hence, 96.

Answer: 445

Explanation :

A triangle can formed by choosing any 3 points  expect all 3 are collinear when we won’t get any triangle.

∴ Total number of triangles = ¹⁵C₃ – ⁵C₃ =  455 – 10 = 445.

Hence, 445.

Answer: 204

Explanation :

Number of squares in a m × n grid = 1² + 2² + 3² + … + n². (where m ≥ n)

∴ Number of squares in this 8 × 8 grid = 1² + 2² + 3² + … + 8² = 204.

Hence, 204.

Answer: 1296

Explanation :

Number of rectangles in a m × n grid = ^m+1C₂ × ⁿ⁺¹C₂.

∴ Number of rectangles in a 8 × 8 grid = ⁹C₂ × ⁹C₂ = 36 × 36 = 1296.

Hence, 1296.

Answer: Option B

Explanation :

To go from A to C, a person has to take 8 steps toward right and 8 steps upwards.

One of the ways is: R R R R R R R R U U U U U U U U 

R → 1 step towards right
U → 1 step upward.

Total number of ways of going from A to C is same as total ways of arranging these 8 identical Rs and 8 identical Us.

∴ Total number of ways = 16!/(8!×8!).

Hence, option (b).

Answer: Option A

Explanation :

Consider the solution to the previous question.

A → B: To go from A to B, we need to take 4 steps towards right and 4 steps upwards i.e., total number of ways = $\frac{8!}{4!\times 4!}$

B → C: To go from B to C, we need to take 4 steps towards right and 4 steps upwards i.e., total number of ways = $\frac{8!}{4!\times 4!}$

A → B → C: To go from A to C via B, total number of ways = $\frac{8!}{4!\times 4!}\times \frac{8!}{4!\times 4!}$ =  $\frac{{\left(8!\right)}^2}{{\left(4!\right)}^4}$.

Hence, option (a).

Answer: Option C

Explanation :

Total number of ways of going from A to C = $\frac{16!}{8!\times 8!}$.

Total number of ways of going from A to C via B = $\frac{{\left(8!\right)}^2}{{\left(4!\right)}^4}$.

Total number of ways of going from A to C without going through B = $\frac{16!}{8!\times 8!}-\frac{{\left(8!\right)}^2}{{\left(4!\right)}^4}$.

Hence, option (c).

Answer: Option B

Explanation :

For n lines in a plane:

Minimum Bounded regions will be 0 when all the lines are parallel to each other.

Minimum Unbounded regions will be (n + 1) when all the lines are parallel to each other.

Hence, option (b).

Answer: 56

Explanation :

Maximum number of regions can be obtained when all lines intersect each other at distinct points i.e., $\frac{n\left(n+1\right)}{2}$+1.

∴ Required answer = $\frac{10\left(10+1\right)}{2}$ + 1 = 56.

Hence, 56.

Answer: Option C

Explanation :

For n lines in a plane:

Maximum Unbounded regions is 2n = 20, when not all lines are parallel.

Maximum Bounded region = n(n+1)/2 + 1 - 2n = 56 – 20 = 36

Hence, option (c).