If A,B,C are subsets of the universal set `U` such that `n(U)=692,n(B)=230,n( C)=370,n(BcapC)=20,n(A' cap B'capC')=10` then: `n(AcapB'capC')=`

Options

A. 876
B. 784
C. 172
D. none of these.

Answer

To solve the problem step by step, we will use the information provided and the properties of sets.

### Step 1: Understand the Given Information
We have the following information:
- \( n(U) = 692 \) (the number of elements in the universal set)
- \( n(B) = 230 \) (the number of elements in set B)
- \( n(C) = 370 \) (the number of elements in set C)
- \( n(B \cap C) = 20 \) (the number of elements in the intersection of sets B and C)
- \( n(A' \cap B' \cap C') = 10 \) (the number of elements in the complement of A, B, and C)

### Step 2: Use the Complement Rule
We know that:
\[ n(A' \cap B' \cap C') = n(U) - n(A \cup B \cup C) \]
Thus, we can write:
\[ n(A \cup B \cup C) = n(U) - n(A' \cap B' \cap C') = 692 - 10 = 682 \]

### Step 3: Apply the Principle of Inclusion-Exclusion
Using the principle of inclusion-exclusion for three sets:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]
Substituting the known values:
\[ 682 = n(A) + 230 + 370 - n(A \cap B) - n(A \cap C) - 20 + n(A \cap B \cap C) \]
This simplifies to:
\[ 682 = n(A) + 600 - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C) \]
Rearranging gives:
\[ n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C) = 82 \quad \text{(Equation 1)} \]

### Step 4: Find \( n(B') \) and \( n(C') \)
Next, we need to find \( n(B') \) and \( n(C') \):
\[ n(B') = n(U) - n(B) = 692 - 230 = 462 \]
\[ n(C') = n(U) - n(C) = 692 - 370 = 322 \]

### Step 5: Calculate \( n(B' \cap C') \)
Using the inclusion-exclusion principle again for \( B' \) and \( C' \):
\[ n(B' \cap C') = n(B') + n(C') - n(B' \cap C) \]
We know:
\[ n(B' \cap C) = n(C) - n(B \cap C) = 370 - 20 = 350 \]
Thus:
\[ n(B' \cap C') = 462 + 322 - 350 = 434 \]

### Step 6: Find \( n(A \cap B' \cap C') \)
We can now find \( n(A \cap B' \cap C') \):
\[ n(A \cap B' \cap C') = n(B' \cap C') - n(A' \cap B' \cap C') \]
Substituting the values:
\[ n(A \cap B' \cap C') = 434 - 10 = 424 \]

### Final Answer
Thus, the number of elements in \( n(A \cap B' \cap C') \) is \( 424 \).