1. **Problem Statement:** We have 100 members in Majawa village who undertake at least one of three activities: going to school (S), farming (F), and trading (T). Given: - $|S|=38$ - $|F|=54$ - $|T|=50$ - $|S \cap T|=18$ - $|F \cap F|=22$ (likely a typo, assume $|F \cap T|=22$) - $|S \cap F|=10$ - Total members $=100$ - Number of people who go to school only equals number who trade only. 2. **Goal:** Find the number of people who undertake all three activities $|S \cap F \cap T|$. 3. **Formula and Rules:** Using the principle of inclusion-exclusion for three sets: $$ |S \cup F \cup T| = |S| + |F| + |T| - |S \cap F| - |F \cap T| - |S \cap T| + |S \cap F \cap T| $$ Since all 100 members undertake at least one activity, $|S \cup F \cup T|=100$. 4. **Define variables:** Let: - $x = |S \cap F \cap T|$ (all three activities) - $a = |S \text{ only}|$ - $b = |F \text{ only}|$ - $c = |T \text{ only}|$ Given $a = c$ (school only equals trade only). 5. **Express intersections excluding triple intersection:** - $|S \cap F| = 10 = a_{SF} + x$ where $a_{SF}$ is people in $S \cap F$ only - $|F \cap T| = 22 = a_{FT} + x$ - $|S \cap T| = 18 = a_{ST} + x$ 6. **Express total counts:** - $|S| = a + a_{SF} + a_{ST} + x = 38$ - $|F| = b + a_{SF} + a_{FT} + x = 54$ - $|T| = c + a_{ST} + a_{FT} + x = 50$ 7. **Total population:** Sum of all disjoint parts equals 100: $$ a + b + c + a_{SF} + a_{FT} + a_{ST} + x = 100 $$ 8. **Use $a = c$ and substitute $a_{SF} = 10 - x$, $a_{FT} = 22 - x$, $a_{ST} = 18 - x$: From $|S|$: $$a + (10 - x) + (18 - x) + x = 38 \Rightarrow a + 28 - x = 38 \Rightarrow a - x = 10$$ From $|F|$: $$b + (10 - x) + (22 - x) + x = 54 \Rightarrow b + 32 - x = 54 \Rightarrow b - x = 22$$ From $|T|$: $$c + (18 - x) + (22 - x) + x = 50 \Rightarrow c + 40 - x = 50 \Rightarrow c - x = 10$$ Since $a = c$, from above: $$a - x = 10$$ $$c - x = 10$$ Both consistent. 9. **From total population:** $$a + b + c + (10 - x) + (22 - x) + (18 - x) + x = 100$$ Substitute $c = a$: $$a + b + a + 10 - x + 22 - x + 18 - x + x = 100$$ $$2a + b + 50 - 2x = 100$$ $$2a + b - 2x = 50$$ 10. **Use equations from step 8 to express $a$ and $b$ in terms of $x$:** $$a = x + 10$$ $$b = x + 22$$ Substitute into step 9: $$2(x + 10) + (x + 22) - 2x = 50$$ $$2x + 20 + x + 22 - 2x = 50$$ $$x + 42 = 50$$ $$x = 8$$ **Answer:** The number of people who undertake all three activities is $\boxed{8}$.