answer:With the inclusion of 'Y', we now have six types of letters (A, E, I, O, U, Y) available for forming words. There are initially 5 choices (A, E, I, O, U) for each position if one does not choose 'Y'. If 'Y' is chosen, since it has a limited appearance (3 times), its availability decreases with each selection. We calculate the total number of combinations as follows: 1. **No 'Y' used**: (5^5) since we can pick any of the 5 vowels for each of the 5 positions. 2. **One 'Y' used**: We can place 'Y' in any one of the 5 positions, and choose freely from the 5 vowels for the other 4 positions. - Combinations = 5 times (5^4) 3. **Two 'Y's used**: We can place 'Y' in any two of the 5 positions, and choose freely from the 5 vowels for the remaining 3 positions. - Combinations = binom{5}{2} times (5^3) 4. **Three 'Y's used**: We can place 'Y' in any three of the 5 positions, and choose freely from the 5 vowels for the remaining 2 positions. - Combinations = binom{5}{3} times (5^2) Calculating each part: - (5^5) = 3125 - 5 times (5^4) = 5 times 625 = 3125 - binom{5}{2} times (5^3) = 10 times 125 = 1250 - binom{5}{3} times (5^2) = 10 times 25 = 250 Adding up all possibilities: Total = 3125 + 3125 + 1250 + 250 = boxed{7750}

answer:Given: - There are 6 problems in a mathematics competition. - Each problem has a score of an integer between 0 and 7, inclusive. - The total score of each participant is the product of the scores of the 6 problems. - If two participants have the same product, the sum of the scores of their problems is considered. - The total number of participants is (8^6 = 262144). - It is mentioned that no two participants have the same scores on all problems. We need to find the score of the participant ranked (7^6 = 117649). 1. **Understanding Total and Unique Scores:** [ text{Total possible combinations of scores} = 8^6 = 262144 ] Each participant has a unique combination of scores for the 6 problems. 2. **Constraints on Scores:** All scores are between 0 and 7. Thus: [ text{Score for each problem, } x_i: 0 leq x_i leq 7 quad text{for } i = 1, 2, ldots, 6 ] 3. **Calculating Unique Products:** If we consider the product of scores: [ P = prod_{i=1}^{6} x_i ] There are (7^6) different non-zero products (product combinations when no score is zero). 4. **Ranking and Identifying the Minimal Non-Zero Product:** To rank the participants: - When participants' products are ranked, the smallest non-zero complete product would be when each problem is scored the minimum non-zero score. - The minimum non-zero score is 1 for each problem. 5. **Validating the Rank and Score:** - The number of valid score combinations where no problem has a score of 0 is (7^6). - The smallest product when none of the scores is zero is achieved with all scores being 1. - Thus, for a participant to have the smallest non-zero product: [ P = 1 cdot 1 cdot 1 cdot 1 cdot 1 cdot 1 = 1 ] Conclusion: [ boxed{1} ]

answer:Instead of calculating a - 2b - 3c, Lily calculated a - 2b + 6c. Therefore, the value of a - 2b can be found adjusting the values: - Correctly, a - 2b - 3c = 2 - Mistakenly, a - 2(b - 3c) = a - 2b + 6c = 14 The system of equations can now be set as: 1. (a - 2b - 3c = 2) 2. (a - 2b + 6c = 14) Subtracting equation 1 from equation 2: [ (a - 2b + 6c) - (a - 2b - 3c) = 14 - 2 ] [ 6c + 3c = 12 ] [ 9c = 12 ] [ c = frac{12}{9} = frac{4}{3} ] Plug the value of c into equation 1: [ a - 2b - 3left(frac{4}{3}right) = 2 ] [ a - 2b - 4 = 2 ] [ a - 2b = 2 + 4 ] [ a - 2b = boxed{6} ] Conclusion: The answer to Lily's problem is boxed{6}, which is consistent with the defined operations and the mistakes she made.

answer:First, we need to convert the speeds of the trains from km/hr to m/s because the lengths of the trains are given in meters. To convert km/hr to m/s, we use the conversion factor: 1 km/hr = 5/18 m/s. Train A's speed in m/s: 54 km/hr * (5/18) = 15 m/s Train B's speed in m/s: 36 km/hr * (5/18) = 10 m/s Since the trains are moving towards each other, we need to add their speeds to find the relative speed at which they are approaching each other. Relative speed = Speed of Train A + Speed of Train B Relative speed = 15 m/s + 10 m/s = 25 m/s Now, we need to find the total distance that Arun will travel to completely cross Train B. This distance is the sum of the lengths of both trains. Total distance = Length of Train A + Length of Train B Total distance = 200 m + 150 m = 350 m Finally, we can calculate the time taken by Arun to completely cross Train B by dividing the total distance by the relative speed. Time taken = Total distance / Relative speed Time taken = 350 m / 25 m/s = 14 seconds So, it will take Arun boxed{14} seconds to completely cross Train B.