(a) Five female and seven male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of empl...
Question 1 Report
(a) Five female and seven male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of employing the 4 teachers, if : (i) there is no restriction ; (ii) at least 2 of them are females.
(b) The table shows the positions awarded to 7 contestants by Judges X and Y in a competition.
| Contestant | P | Q | R | S | T | U | V |
| Judge X | 2 | 7 | 1 | 3 | 6 | 5 | 4 |
| Judge Y | 4 | 6 | 2 | 3 | 1 | 1 | 5 |
(i) Calculate, correct to one decimal place, the Spearman's rank correlation coefficient.
(ii) Interpret your answer in b(i) above.
Answer Details
(a) Selecting 4 teachers from 5 female + 7 male = 12 teachers.
(i) No restriction. Choose any 4 of the 12:
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \mathbf{495 \text{ ways}} \](ii) At least 2 females. The cases are exactly 2, 3 or 4 females (the rest male):
\[ \binom{5}{2}\binom{7}{2} + \binom{5}{3}\binom{7}{1} + \binom{5}{4}\binom{7}{0} \] \[ = (10)(21) + (10)(7) + (5)(1) = 210 + 70 + 5 = \mathbf{285 \text{ ways}} \](b) Spearman's rank correlation for 7 contestants (\(n = 7\)). Judge Y has a tie (T and U both awarded position 1), so they share positions 1 and 2, each taking the average rank \(1.5\); the remaining Y values are then re-ranked \(2\to3, 3\to4, 4\to5, 5\to6, 6\to7\). Judge X already has distinct positions 1 to 7.
| Contestant | \(R_X\) | \(R_Y\) | \(d\) | \(d^2\) |
|---|---|---|---|---|
| P | 2 | 5 | -3 | 9 |
| Q | 7 | 7 | 0 | 0 |
| R | 1 | 3 | -2 | 4 |
| S | 3 | 4 | -1 | 1 |
| T | 6 | 1.5 | 4.5 | 20.25 |
| U | 5 | 1.5 | 3.5 | 12.25 |
| V | 4 | 6 | -2 | 4 |
| Total \(\sum d^2\) | 50.5 | |||
(ii) Interpretation. \(r_s \approx 0.1\) is very close to zero, so there is only a very weak positive correlation. The two judges show almost no agreement in the way they ranked the contestants.
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