Explanation

When three 6-faced dice are thrown together, the total number of possible outcomes is 6 × 6 × 6 = 216 (since each die has 6 possible outcomes).

The favorable outcomes where all three dice show the same number are:

(1, 1, 1), (2, 2, 2), (3, 3, 3), (4, 4, 4), (5, 5, 5), and (6, 6, 6)

There are 6 favorable outcomes.

The probability of all three dice showing the same number is:

Number of favorable outcomes / Total number of possible outcomes

= 6 / 216

= 1/36

Explanation

A leap year has 366 days, which means 52 full weeks (364 days) and 2 extra days. The extra days can be:

• Sunday and Monday
• Monday and Tuesday, etc.

For a leap year to have 53 Sundays and 52 Mondays, the two extra days must be Sunday and Monday, which happens in 2 out of 7 possible combinations.

Hence, the probability is 2/7.

Explanation

To find the probability that a pair of books will always be together, we need to count the number of arrangements where the pair is together and divide it by the total number of arrangements.

Let's consider the pair of books as a single unit. Then, we have 9 units (the pair and the remaining 8 books) to arrange on the shelf. The number of ways to arrange these 9 units is 9!.

However, within the pair, the two books can be arranged in 2! ways (either book can be on the left or right).

So, the total number of arrangements where the pair is together is 9! × 2!.

Now, we divide the number of favorable arrangements (where the pair is together) by the total number of arrangements:

Probability = (9! × 2!) / 10!

= (9! / 10!) × 2

= (1/10) × 2

= 1/5

Explanation

The mode is the digit that appears most frequently in the data set.

Let's count the occurrences of each digit:

• 0: 3 times
• 1: 2 times
• 2: 1 time
• 3: 2 times
• 5: 1 time
• 6: 2 times
• 7: 1 time
• 9: 1 time

The digit 0 appears most frequently, 3 times. Therefore, the mode is 0.

Explanation

In an arithmetic progression (AP), the nth term is given by the formula

Tn=a+(n−1)⋅d

Where:

a is the first term

d is the common difference,

n is the term number.

For the given AP:

First term a=2,

Common difference  d=11−2=9,

We are looking for  n when the term is 290.

Let the given AP contain n terms.

Tn = 290

=> a + (n – 1)d = 290

=> 2 + (n – 1)9 = 290

=> 2 + 9n – 9 = 290

=> 9n = 297

=> n = 33

Thus, the 33 term of the AP is 290.

Explanation

The equation $text{IQ} = frac{text{Mental Age (M.A.)}}{text{Chronological Age (C.A.)}} times 100$ was developed by Alfred Binet and Theodore Simon.

This formula was part of the Binet-Simon scale, designed to measure intelligence in children.

It was later adapted and popularized in various forms, such as the Stanford-Binet IQ test.

Explanation

Total number of balls = 6 black + 8 white = 14 balls

Number of white balls = 8

Probability of drawing a white ball = Number of white balls / Total number of balls

= 8/14

= 4/7

So, the probability of drawing a white ball is 4/7.

Explanation

From the above-shown logic tables, we can say that for the NAND and NOR gate, we will get the output '0' when both the inputs are 1.

Explanation

• To find the mean of a set of numbers, you add up all the numbers and then divide by the total count of numbers.
• For the scores 1, 2, 3, 4, 5:
• (1 + 2 + 3 + 4 + 5) / 5 = 15 / 5 = 3
• So, the mean of the scores is 3.

Explanation

To find the interest earned using the simple interest formula, use:

Simple Interest = 𝑃 × 𝑟 × 𝑡

where:

P is the principal amount ($1200),

r is the annual interest rate (5% or 0.05),

t is the time in years (2 years).

Substitute the values into the formula:

Simple Interest=1200×0.05×2

Calculate:

Simple Interest=1200×0.05=60

Simple Interest=60×2=120