Let A, B, C be subsets of the universal set U. If `n(U)=692,n(B)=230,
n(C)=370,n(BnnC)=20,n(AnnB'nnC')=10`, find `n(A'nnB'nnC')`.

Answer

To solve the problem, we need to find \( n(A' \cap B' \cap C') \) given the following information:

- \( n(U) = 692 \)
- \( n(B) = 230 \)
- \( n(C) = 370 \)
- \( n(B \cap C) = 20 \)
- \( n(A \cap B' \cap C') = 10 \)

### Step 1: Find \( n(B \cup C) \)

Using the principle of inclusion-exclusion for two sets, we have:

\[ n(B \cup C) = n(B) + n(C) - n(B \cap C) \]

Substituting the values:

\[ n(B \cup C) = 230 + 370 - 20 = 580 \]

### Step 2: Find \( n(A' \cap B' \cap C') \)

We know that the total number of elements in the universal set \( U \) is 692. The number of elements in \( A' \cap B' \cap C' \) can be calculated as:

\[ n(A' \cap B' \cap C') = n(U) - n(B \cup C) - n(A \cap B' \cap C') \]

Substituting the known values:

\[ n(A' \cap B' \cap C') = 692 - 580 - 10 \]

Calculating this gives:

\[ n(A' \cap B' \cap C') = 692 - 580 - 10 = 102 \]

### Final Answer

Thus, the number of elements in \( A' \cap B' \cap C' \) is:

\[ \boxed{102} \]