Explanation

Let's evaluate the expressions using the given definition of a * b:

3 * 4 = 3 + 4 + (3)(4) = 3 + 4 + 12 = 19

2 * 3 = 2 + 3 + (2)(3) = 2 + 3 + 6 = 11

Now, let's subtract:

3 * 4 - 2 * 3 = 19 - 11 = 8

Explanation

A polygon with 8 sides has 8 vertices. To form a triangle, you need to choose 3 vertices out of these 8. This can be done in:

8C3 = 8! / (3! x (8-3)!)

= 8! / (3! x 5!)

= (8 x 7 x 6) / (3 x 2 x 1)

= 56

So, 56 triangles can be formed by joining the angular points of a polygon with 8 sides.

Explanation

The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours.

So, in a day, the hands point in the opposite directions 22 times.

Explanation

The ratio of the length of the rod to its shadow is 1:√3.

This ratio is equal to the tangent of the angle of elevation of the sun (tan(θ) = opposite/adjacent).

So, we can set up the equation: tan(θ) = 1/√3

To find the angle θ, we take the inverse tangent (arctangent) of both sides: θ = arctan(1/√3)

Using a calculator or trigonometric tables, we find that θ = 30°

Explanation

The number of ways to choose 2 students from a class of 20 students is represented by the combination formula:

C(20,2) = 20! / (2! x (20-2)!)

where ! denotes the factorial function.

Simplifying the formula, we get:

C(20,2) = (20 x 19) / 2

= 190

So, there are 190 ways to choose 2 students from a class of 20 students.

Explanation

After 10 days: 150 men had food for 35 days.

Suppose 125 men had food for x days.

Now, Less men, More days (Indirect Proportion)

 125 : 150 :: 35 : x        125 x x = 150 x 35

 x =         150 x 35

125

 x = 42.

Explanation

To find the percentage of boys, divide the number of boys (56) by the total number of students (140), and then multiply by 100:

(56 ÷ 140) x 100 = 0.4 x 100 = 40%

So, the percentage of boys is 40%.

Explanation

Average is equal to sum of all numbers and divided by Total numbers

= 9+18+27+36+45 = 135/5

=27

Explanation

1/0.02= 50

Explanation

The innings in cricket is always plural so 3*12= 36 80-36= 44 so his previous average was 44.

Now after 12th innings his average is 44+3= 47