Combining multiple average relations: Average of (a, b, c) is 11; average of (c, d, e) is 17; average of (e, f) is 22; and average of (e, c) is 17. What is the average of (a, b, c, d, e, f)?

Introduction / Context:
We are given several overlapping averages and asked for the combined average. Eliminating variables through sums is effective.

Given Data / Assumptions:

• (a + b + c) / 3 = 11 → a + b + c = 33
• (c + d + e) / 3 = 17 → c + d + e = 51
• (e + f) / 2 = 22 → e + f = 44
• (e + c) / 2 = 17 → e + c = 34

Concept / Approach:
Compute S = a + b + c + d + e + f using the equations. Note that (c + d + e) + (a + b + c) + (e + f) - (e + c) subtracts the overlap to yield S.

Step-by-Step Solution:
S = (a + b + c) + (c + d + e) + (e + f) - (e + c)S = 33 + 51 + 44 - 34 = 94Average of six = S / 6 = 94 / 6 = 47 / 3 = 15 2/3

Verification / Alternative check:
Express variables in terms of parameters and show total remains 94 regardless, confirming invariance.

Why Other Options Are Wrong:
151⁄3, 162⁄3, 171⁄2 are not equal to 47/3; only 152⁄3 matches 15 2/3.

Common Pitfalls:
Double-counting c and e; forgetting to subtract the shared pair once.

Final Answer:
152⁄3