4
We are given a problem where we need to find the arithmetic mean of five numbers: A, B, C, D, and E. The sum of all possible groups of 3 distinct numbers from these five numbers is 120. Let's break down the solution step-by-step:
- First, determine the total number of groups. The combination of selecting 3 numbers out of 5 is given by \(\binom{5}{3}\) which equates to: \(5 \times 4 \times 3 / (3 \times 2 \times 1) = 10\) groups.
- Since the problem states that the aggregate (total) of the sums of numbers of each group is 120, we can conclude that the sum of all these groups' totals is 120.
- Note that each number (A, B, C, D, and E) appears in exactly \(\binom{4}{2} = 6\) different 3-number groups (as each is paired with two other numbers).
- The total sum of all elements in all groups is thus effectively \(6(A + B + C + D + E) = 120\).
- Divide both sides by 6 to find the sum of A, B, C, D, and E: \(A + B + C + D + E = \frac{120}{6} = 20\).
- Finally, calculate the arithmetic mean by dividing the total sum of the numbers by 5 (since there are 5 numbers): \(\text{Arithmetic Mean} = \frac{A + B + C + D + E}{5} = \frac{20}{5} = 4\).
Therefore, the arithmetic mean of A, B, C, D, and E is 4. This matches the given option.
Important Questions from Average
The average height of 20 students of class 8 is 152 cm and the average height of 15 students of class 9 is 168 cm. What is the average height (to the nearest cm) of the students of both classes?
The average of 4, 6, 8, 12 and x is 7 and the average of x, 9, 13, 15 and y is 9. What is the value of 2x - 3y?
The average weight of 20 girls in a school was 52 kg. Two new students of weight 54 kg and 50 kg were admitted. The ratio of this new average to the old one is:
If the average of two numbers is 13 and the square root of their product is 12, then the difference between the numbers is:
If the average of 5 consecutive odd integers in increasing order is 11 , then the average of the last 3 of them is:
