This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back — been a while! Let's pick up where you left off. The research team has: • 8 Software engineers • 6 Analysts • 4 Programmers Total members = 8 + 6 + 4 = 18. A committee of 4 members is to be selected. We use the combination formula C(n, k) = (n!)/(k!(n-k)!). i) All Software engineers This means all 4 members selected must be Software engineers. There are 8 Software engineers available. Step 1: Calculate the number of ways to choose 4 Software engineers from 8. C(8, 4) = (8!)/(4!(8-4)!) = (8!)/(4!4!) = (8 × 7 × 6 × 5)/(4 × 3 × 2 × 1) = (1680)/(24) = 70 The number of committees consisting of all Software engineers is 70. ii) At least two are analysts This means the committee can have 2, 3, or 4 analysts. The remaining members will be chosen from the non-analysts (8 Software engineers + 4 Programmers = 12 non-analysts). Step 1: Calculate for 2 analysts. Choose 2 analysts from 6: C(6, 2) = (6 × 5)/(2 × 1) = 15. Choose 2 non-analysts from 12: C(12, 2) = (12 × 11)/(2 × 1) = 66. Number of ways = 15 × 66 = 990. Step 2: Calculate for 3 analysts. Choose 3 analysts from 6: C(6, 3) = (6 × 5 × 4)/(3 × 2 × 1) = 20. Choose 1 non-analyst from 12: C(12, 1) = 12. Number of ways = 20 × 12 = 240. Step 3: Calculate for 4 analysts. Choose 4 analysts from 6: C(6, 4) = (6 × 5 × 4 × 3)/(4 × 3 × 2 × 1) = 15. Choose 0 non-analysts from 12: C(12, 0) = 1. Number of ways = 15 × 1 = 15. Step 4: Sum the possibilities. Total ways = 990 + 240 + 15 = 1245. The number of committees with at least two analysts is 1245. iii) At most one programmer is among the 4 This means the committee can have 0 or 1 programmer. The remaining members will be chosen from the non-programmers (8 Software engineers + 6 Analysts = 14 non-programmers). Step 1: Calculate for 0 programmers. Choose 0 programmers from 4: C(4, 0) = 1. Choose 4 non-programmers from 14: C(14, 4) = (14 × 13 × 12 × 11)/(4 × 3 × 2 × 1) = 1001. Number of ways = 1 × 1001 = 1001. Step 2: Calculate for 1 programmer. Choose 1 programmer from 4: C(4, 1) = 4. Choose 3 non-programmers from 14: C(14, 3) = (14 × 13 × 12)/(3 × 2 × 1) = 364. Number of ways = 4 × 364 = 1456. Step 3: Sum the possibilities. Total ways = 1001 + 1456 = 2457. The number of committees with at most one programmer is 2457. Send me the next one 📸