(a) A population of 120 individuals contains 45 engineers, 30 teachers, and the rest are administrators. A simple random sample of 6 is selected. (i) Probability that exactly 3 engineers are included? (ii) Probability that the sample contains no teachers? (b) A length is measured five times: 12.478 m, 12.463 m, 12.481 m, 12.469 m, 12.472 m. Determine the absolute error between the exact average and the rounded average. (c) Evaluate the argument "Artificial Intelligence will reduce employment because automation replaces human labor." (d) A number leaves remainder 5 when divided by 12 and remainder 4 when divided by 7. Find the smallest such positive integer and its remainder when divided by 9.

(a) Probability

Population: 45 engineers, 30 teachers, 120 − 75 = 45 administrators. Since the sample is drawn without replacement, use combinations.

Total ways of choosing 6 from 120: C(120, 6) = (120 × 119 × 118 × 117 × 116 × 115) / 720 = 3,652,745,460.

(i) Exactly 3 engineers: choose 3 of the 45 engineers and 3 of the other 75 people.

C(45, 3) = 14,190 and C(75, 3) = 67,525, so favourable outcomes = 14,190 × 67,525 = 958,179,750.

P = 958,179,750 / 3,652,745,460 ≈ 0.262 (about 26.2%).

(ii) No teachers: all 6 come from the 90 non-teachers.

C(90, 6) = (90 × 89 × 88 × 87 × 86 × 85) / 720 = 622,614,630.

P = 622,614,630 / 3,652,745,460 ≈ 0.170 (about 17.0%).

(b) Absolute error

Sum = 12.478 + 12.463 + 12.481 + 12.469 + 12.472 = 62.363 m.

Exact average = 62.363 / 5 = 12.4726 m.

The data are recorded to three decimal places, so the rounded average = 12.473 m.

Absolute error = |12.473 − 12.4726| = 0.0004 m (0.4 mm).

(c) Evaluating the AI argument

Assumptions: (1) the tasks AI automates make up whole jobs, not parts of jobs; (2) the economy has a fixed amount of work (the "lump of labour" idea); (3) workers cannot be retrained; (4) productivity gains will not create new demand or new industries.

Evaluation: the argument is weak as a general claim but has some force in the short run. It is true that automation displaces specific tasks — data entry, routine customer service, some clerical and manufacturing work — and that displaced workers can suffer. But the conclusion that total employment will fall does not follow from the premise.

Counter-argument: history shows that technology both destroys and creates jobs. Mechanisation of agriculture and computerisation did not cause permanent mass unemployment; they shifted work to new sectors. AI lowers costs and raises productivity, which increases demand and creates new roles (data annotation, AI maintenance, cybersecurity, digital services). Many jobs will be augmented rather than replaced — doctors, teachers and lawyers using AI tools. The real effect depends on policy: education, reskilling and social protection. A more accurate claim would be: "AI will change the structure of employment and may displace certain groups of workers unless retraining and policy keep pace."

(d) Remainders

Numbers that leave remainder 5 when divided by 12: 5, 17, 29, 41, 53, 65, …

Check each when divided by 7: 5 → 5, 17 → 3, 29 → 1, 41 → 6, 53 → 4 ✓.

So the smallest such integer is N = 53 (all solutions are 53 + 84k, since 12 × 7 = 84).

Remainder when 53 is divided by 9: 53 = 9 × 5 + 8, so the remainder is 8.