1. **Problem Q1: Probability with cards drawn without replacement** (a) Both cards are red. - Total cards = 52, red cards = 26. - Probability first card red = $\frac{26}{52} = \frac{1}{2}$. - After drawing one red card, red cards left = 25, total cards left = 51. - Probability second card red = $\frac{25}{51}$. - Combined probability = $\frac{1}{2} \times \frac{25}{51} = \frac{25}{102}$. (b) First card black, second red. - Black cards = 26. - Probability first card black = $\frac{26}{52} = \frac{1}{2}$. - After drawing black card, red cards still 26, total cards left 51. - Probability second card red = $\frac{26}{51}$. - Combined probability = $\frac{1}{2} \times \frac{26}{51} = \frac{26}{102} = \frac{13}{51}$. (c) Both cards are aces. - Total aces = 4. - Probability first ace = $\frac{4}{52} = \frac{1}{13}$. - After drawing one ace, aces left = 3, total cards left = 51. - Probability second ace = $\frac{3}{51} = \frac{1}{17}$. - Combined probability = $\frac{1}{13} \times \frac{1}{17} = \frac{1}{221}$. 2. **Problem Q2: Number of 4-digit numbers from digits 1-6 without repetition** - Digits available: 6. - Number of 4-digit numbers without repetition = permutations of 6 digits taken 4 at a time. - Formula: $P(n,r) = \frac{n!}{(n-r)!}$. - Calculation: $\frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{720}{2} = 360$. 3. **Problem Q3: Number of 5-letter words from LEADER** - Letters: L, E, A, D, E, R (6 letters, E repeated twice). - Number of 5-letter words = number of permutations of 6 letters taken 5 at a time with repetition of E considered. - Total permutations of 6 letters with 2 E's: $\frac{6!}{2!} = 360$. - For 5-letter words, consider cases: - Case 1: Both E's included. - Choose 3 letters from {L, A, D, R} = $\binom{4}{3} = 4$. - Number of arrangements = $\frac{5!}{2!} = 60$. - Total = $4 \times 60 = 240$. - Case 2: Only one E included. - Choose 4 letters from {L, A, D, R} = $\binom{4}{4} = 1$. - Number of arrangements = $5! = 120$. - Total = $1 \times 120 = 120$. - Total 5-letter words = $240 + 120 = 360$. 4. **Problem Q4: Seating 6 students in a row** (a) Two particular students must sit together. - Treat the two students as one unit. - Number of units = 5 (the pair + 4 others). - Number of arrangements = $5! = 120$. - The two students can switch seats: $2! = 2$ ways. - Total arrangements = $120 \times 2 = 240$. (b) Two particular students must not sit together. - Total arrangements without restriction = $6! = 720$. - Subtract arrangements where they sit together = 240. - Result = $720 - 240 = 480$. 5. **Problem Q5: Two fair dice rolled** - Total outcomes = $6 \times 6 = 36$. (a) Probability sum is 8. - Possible pairs: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 outcomes. - Probability = $\frac{5}{36}$. (b) Probability sum is even. - Sum even if both dice are even or both odd. - Even numbers: 2,4,6 (3 numbers), odd numbers: 1,3,5 (3 numbers). - Even sum outcomes = $3 \times 3 + 3 \times 3 = 18$. - Probability = $\frac{18}{36} = \frac{1}{2}$. (c) Probability sum greater than 9. - Sums > 9: 10, 11, 12. - Sum 10: (4,6), (5,5), (6,4) = 3. - Sum 11: (5,6), (6,5) = 2. - Sum 12: (6,6) = 1. - Total = 6. - Probability = $\frac{6}{36} = \frac{1}{6}$. (d) Probability sum is prime. - Prime sums possible: 2,3,5,7,11. - Sum 2: (1,1) = 1. - Sum 3: (1,2), (2,1) = 2. - Sum 5: (1,4), (2,3), (3,2), (4,1) = 4. - Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6. - Sum 11: (5,6), (6,5) = 2. - Total = 1+2+4+6+2 = 15. - Probability = $\frac{15}{36} = \frac{5}{12}$. 6. **Problem Q6: Gain or loss percentage** - Cost price = 15000 + 3500 = 18500. - Selling price = 18200. - Loss = 18500 - 18200 = 300. - Loss percentage = $\frac{300}{18500} \times 100 = 1.62\%$ loss. 7. **Problem Q7: Cost price from selling price and gain percentage** - Selling price = 1280. - Gain = 7.5%. - Cost price = $\frac{Selling\ Price}{1 + \frac{Gain}{100}} = \frac{1280}{1.075} \approx 1190.70$. 8. **Problem Q8: Profit percentage on chair** - Cost price = 900 + 150 = 1050. - Selling price = 1400. - Profit = 1400 - 1050 = 350. - Profit percentage = $\frac{350}{1050} \times 100 = 33.33\%$.