To solve the problem step by step, we will use the information given about the averages of different groups of scores.
### Step 1: Calculate the total scores of A, B, and C
The average score of A, B, and C is 78. Therefore, we can express their total score as:
\[
A + B + C = \text{Average} \times \text{Number of Students} = 78 \times 3 = 234
\]
### Step 2: Calculate the total scores of C, D, and E
The average score of C, D, and E is 52. Thus, we can express their total score as:
\[
C + D + E = 52 \times 3 = 156
\]
### Step 3: Calculate the total scores of E and F
The average score of E and F is 48. Therefore, we can express their total score as:
\[
E + F = 48 \times 2 = 96
\]
### Step 4: Calculate the total scores of E and C
The average score of E and C is 60. Thus, we can express their total score as:
\[
E + C = 60 \times 2 = 120
\]
### Step 5: Solve for D
From the equation \(C + D + E = 156\) and \(E + C = 120\), we can find D:
\[
D = (C + D + E) - (E + C) = 156 - 120 = 36
\]
### Step 6: Solve for E
Now we can substitute D back into the equation for \(E + F\):
\[
E + F = 96
\]
We already have \(E + C = 120\), so we can find E:
\[
E = 120 - C
\]
Substituting \(E\) into \(E + F = 96\):
\[
(120 - C) + F = 96 \implies F = 96 - (120 - C) = C - 24
\]
### Step 7: Find the total of D, E, and F
Now we can find the total of D, E, and F:
\[
D + E + F = 36 + E + F = 36 + (120 - C) + (C - 24) = 36 + 120 - 24 = 132
\]
### Step 8: Calculate the total score of A, B, C, D, E, and F
Now we can find the total score of A, B, C, D, E, and F:
\[
\text{Total} = (A + B + C) + (D + E + F) = 234 + 132 = 366
\]
### Step 9: Calculate the average score of A, B, C, D, E, and F
Finally, we calculate the average score of all six individuals:
\[
\text{Average} = \frac{\text{Total}}{\text{Number of Students}} = \frac{366}{6} = 61
\]
### Final Answer
The average score of A, B, C, D, E, and F is **61**.
