1. **Problem 1: Clock Angle at 5:42 pm** We want to find the angle formed by the hour hand to the minute hand in the clockwise direction at 5:42 pm. 2. **Formula and rules:** - The minute hand moves 6 degrees per minute: $$6 \times \text{minutes}$$ - The hour hand moves 30 degrees per hour plus 0.5 degrees per minute: $$30 \times \text{hours} + 0.5 \times \text{minutes}$$ - The angle between the hands is the absolute difference between their positions. 3. **Calculate positions:** - Minute hand position: $$6 \times 42 = 252^\circ$$ - Hour hand position: $$30 \times 5 + 0.5 \times 42 = 150 + 21 = 171^\circ$$ 4. **Find clockwise angle from hour to minute hand:** - Difference: $$252 - 171 = 81^\circ$$ - Since this is positive and less than 360, the clockwise angle is $$81^\circ$$. --- 5. **Problem 2: Number of words from letters I, I, I, S, S, T, V** We want to find how many distinct arrangements (words) can be formed from the letters: 3 I's, 2 S's, 1 T, and 1 V. 6. **Formula and rules:** - Total letters: $$7$$ - Number of arrangements with repeated letters: $$\frac{7!}{3! \times 2! \times 1! \times 1!}$$ 7. **Calculate factorials:** - $$7! = 5040$$ - $$3! = 6$$ - $$2! = 2$$ 8. **Calculate number of words:** $$\frac{5040}{6 \times 2} = \frac{5040}{12} = 420$$ **Final answers:** - Clock angle: $$81^\circ$$ - Number of words: $$420$$