1. **Stating the problem:** We have three sets representing professionals working in the morning (M), afternoon (A), and night (N). Given the total counts for each set and their intersections, we want to find: a. The number of professionals working exactly two shifts. b. The number of professionals working exactly one shift. c. The total number interviewed. 2. **Given data:** - $|M|=3224$ - $|A|=3571$ - $|N|=5656$ - $|M \cap A|=1820$ - $|A \cap N|=2467$ - $|M \cap A \cap N|=1545$ 3. **Finding the number working exactly two shifts:** - Those working exactly in $M$ and $A$, but not $N$ is $|M \cap A| - |M \cap A \cap N| = 1820 - 1545 = 275$ - Those working exactly in $A$ and $N$ but not $M$ is $|A \cap N| - |M \cap A \cap N| = 2467 - 1545 = 922$ - We are not given $|M \cap N|$ directly, so to find it, use the inclusion-exclusion principle later, but since no data for this intersection is provided, assume it equals $x$ temporarily. 4. **To find $|M \cap N|$ we use the inclusion-exclusion for total interviewed later. For now, we note that the number working exactly two shifts is sum of the exactly two intersections: $$ \text{Exactly two} = (|M \cap A| - |M \cap A \cap N|) + (|A \cap N| - |M \cap A \cap N|) + (|M \cap N| - |M \cap A \cap N|) $$ This equals: $$ 275 + 922 + (|M \cap N| - 1545) $$ 5. **Finding $|M \cap N|$:** By inclusion-exclusion for total interviewed $T$: $$ T = |M| + |A| + |N| - |M \cap A| - |A \cap N| - |M \cap N| + |M \cap A \cap N| $$ Rearranged to solve for $|M \cap N|$: $$ |M \cap N| = |M| + |A| + |N| - |M \cap A| - |A \cap N| + |M \cap A \cap N| - T $$ We will find $T$ after finding the exact numbers in each set. 6. **Find the number working exactly one shift:** These are those in each set minus those working in overlap: - Exactly $M$ only: $$ |M| - |M \cap A| - |M \cap N| + |M \cap A \cap N| $$ - Exactly $A$ only: $$ |A| - |M \cap A| - |A \cap N| + |M \cap A \cap N| $$ - Exactly $N$ only: $$ |N| - |A \cap N| - |M \cap N| + |M \cap A \cap N| $$ 7. **Express total $T$ as sum of exactly one, exactly two, and exactly three:** $$ T = ( ext{exactly one}) + ( ext{exactly two}) + |M \cap A \cap N| $$ 8. **Calculate $|M \cap N|$ numerically:** We rearrange terms and solve with assumed $T$ as the sum of individuals. Since this is unknown, but consistent with the data, calculate $T$ by adding the known parts. Using the general formula: $$ T = |M| + |A| + |N| - |M \cap A| - |A \cap N| - |M \cap N| + |M \cap A \cap N| $$ We observe from the problem wording, total interviewed should be consistent with these counts. Assuming no further data, we solve for $|M \cap N|$ by assuming no data missing: Let us rearrange for $|M \cap N|$: $$ |M \cap N| = |M| + |A| + |N| - |M \cap A| - |A \cap N| + |M \cap A \cap N| - T $$ For $T$: Since all professionals were interviewed, total $T$ is number to find. We can also express $T$ as sum of all exactly one, exactly two, exactly three. Exact calculations: - Using formula for total unique elements in three sets: $$ T = |M| + |A| + |N| - |M \cap A| - |A \cap N| - |M \cap N| + |M \cap A \cap N| $$ Substituting known numbers: $$ T = 3224 + 3571 + 5656 - 1820 - 2467 - |M \cap N| + 1545 $$ Simplify: $$ T = 12451 - 4287 - |M \cap N| + 1545 = 12451 - 4287 + 1545 - |M \cap N| = 12451 - 2742 - |M \cap N| = 9709 - |M \cap N| $$ But $T$ cannot be expressed without $|M \cap N|$. To find $|M \cap N|$, use totals in sets. Since the problem lacks direct data for $|M \cap N|$, consider total interviewed reported or can be found by logic. 9. **Approximate $|M \cap N|$ using inclusion-exclusion:** Sum the professional numbers: Morning + Afternoon + Night = 3224 + 3571 + 5656 = 12451 Subtract double counted: $|M \cap A| + |A \cap N| + |M \cap N| = 1820 + 2467 + |M \cap N|$ Add $|M \cap A \cap N| = 1545$ So: $$ T = 12451 - (1820 + 2467 + |M \cap N|) + 1545 = 12451 - 4287 - |M \cap N| + 1545 = (12451 + 1545) - 4287 - |M \cap N| = 13996 - 4287 - |M \cap N| = 9709 - |M \cap N| $$ Also $T$ must be at least the largest set size (5656) and likely less than 12451, so for $T$ to be minimal, $|M \cap N|$ should be estimated. Check values: If $T=$9600, then $$ 9600 = 9709 - |M \cap N| \implies |M \cap N| = 109 $$ If so, - Exactly $M$ and $N$ but not $A$: $$ |M \cap N| - |M \cap A \cap N| = 109 - 1545 = -1436 $$ Negative, impossible. Therefore, this suggests $|M \cap N| \,\geq\, 1545$ Assuming the triple intersection $|M \cap A \cap N|=1545$ is contained within $|M \cap N|$, select $|M \cap N| = 1545$ (minimum possible). Hence: - Exactly $M$ and $N$ only: $$ 1545 -1545=0 $$ 10. **Compute exactly two shifts workers:** $$ 275 + 922 + 0 = 1197 $$ 11. **Compute exactly one shift workers:** - Exactly $M$: $$ 3224 - 1820 - 1545 + 1545 = 3224 -1820 = 1404 $$ - Exactly $A$: $$ 3571 - 1820 - 2467 + 1545 = 3571 - 4287 + 1545 = (3571 + 1545) - 4287 = 5116 - 4287 = 829 $$ - Exactly $N$: $$ 5656 - 2467 - 1545 + 1545 = 5656 - 2467 = 3189 $$ Sum exactly one shift: $$ 1404 + 829 + 3189 = 5422 $$ 12. **Total interviewed $T$:** $$ T = ext{exactly one} + ext{exactly two} + ext{exactly three} = 5422 + 1197 + 1545 = 8164 $$ **Final answers:** \a. Number working exactly two shifts: $\boxed{1197}$ \b. Number working exactly one shift: $\boxed{5422}$ \c. Total number interviewed: $\boxed{8164}$