answer:First, simplify the factorials using the identities n! = n cdot (n-1)! and (n+1)! = (n+1) cdot n!: begin{align*} frac{10! cdot 7! cdot 3!}{9! cdot 8!} &= frac{10 cdot 9! cdot 7! cdot 3 cdot 2 cdot 1}{9! cdot 8 cdot 7!} &= frac{10 cdot 7! cdot 6}{8 cdot 7!} &= frac{10 cdot 6}{8} &= frac{60}{8} &= boxed{7.5}. end{align*}

answer:Let's call the number of cookies Paco ate initially "x". After eating some cookies, Paco had 25 - x cookies left. Then he bought 3 more cookies, so he had (25 - x) + 3 cookies. He ate 2 more cookies than those he bought, so he ate 3 + 2 cookies, which equals 5 cookies. Now, we add the cookies he ate initially (x) to the cookies he ate later (5) to find the total number of cookies Paco ate: Total cookies eaten = x + 5 Since we don't have the exact number of cookies Paco ate initially, we can't determine the exact total number of cookies he ate. However, we know that he ate 5 cookies after buying more, so the minimum number of cookies he ate is boxed{5} . If you need the exact total, you would need to provide the number of cookies Paco ate initially.

answer:1. Let the value of the middle pearl be (x) forints. 2. The total number of pearls is 33. 3. We split the pearls into two groups, excluding the middle pearl. Each group will have 16 pearls. We need to calculate the total value of the pearls using the given information about how the value changes towards each end. 4. On one side of the middle pearl, the value decreases by 4500 forints for each pearl. On the other side of the middle pearl, the value decreases by 3000 forints for each pearl. Let's denote the middle pearl's value as (x). We will calculate the sum of the values for each half of the necklace and then sum them together with the value of the middle pearl. **Calculation of one side (decreasing by 4500 forints each pearl):** 5. The values of the pearls on this side can be represented as: [ (x-4500), (x-2 cdot 4500), (x-3 cdot 4500), ldots, (x - 16 cdot 4500) ] This sequence is an arithmetic progression with: [ a_1 = x - 4500 ] [ a_{16} = x - 16 cdot 4500 ] 6. The sum of the first 16 terms of an arithmetic progression is given by: [ S_{16} = frac{16}{2} left[ 2a_1 + (16-1)d right] ] where (d = -4500). Substituting in our values: [ S_{16} = frac{16}{2} left[ 2(x-4500) + 15(-4500) right] ] [ S_{16} = 8 left[ 2(x - 4500) - 67500 right] ] [ S_{16} = 8 left[ 2x - 9000 - 67500 right] ] [ S_{16} = 8 left[ 2x - 76500 right] ] [ S_{16} = 16x - 612000 ] **Calculation of the other side (decreasing by 3000 forints each pearl):** 7. The values of the pearls on the other side can be represented as: [ (x-3000), (x-2 cdot 3000), (x-3 cdot 3000), ldots, (x - 16 cdot 3000) ] This sequence is an arithmetic progression with: [ a_1 = x - 3000 ] [ a_{16} = x - 16 cdot 3000 ] 8. The sum of the first 16 terms of this arithmetic progression is: [ S_{16} = frac{16}{2} left[ 2a_1 + (16-1)d right] ] where (d = -3000). Substituting in our values: [ S_{16} = frac{16}{2} left[ 2(x-3000) + 15(-3000) right] ] [ S_{16} = 8 left[ 2(x - 3000) - 45000 right] ] [ S_{16} = 8 left[ 2x - 6000 - 45000 right] ] [ S_{16} = 8 left[ 2x - 51000 right] ] [ S_{16} = 16x - 408000 ] 9. Summing all the pearls’ values together: [ Total Value = x + 16x - 612000 + 16x - 408000 ] [ Total Value = 33x - 1020000 ] 10. According to the problem, the total value of the pearl necklace is 25 times the value of the fourth pearl from the center on the more valuable side. 11. The fourth pearl from the center on the 3000 forints decrement side: [ (x - 3000) - 3 cdot 3000 = x - 12000 ] 12. According to the problem: [ 33x - 1020000 = 25(x - 12000) ] 13. Solving the equation: [ 33x - 1020000 = 25x - 300000 ] [ 33x - 25x = 1020000 - 300000 ] [ 8x = 720000 ] [ x = frac{720000}{8} = 90000 ] Conclusion: The value of the middle pearl is (boxed{90000}) forints.

answer:1. **Determine the normal price of 5 tickets:** Let's denote the normal price of a single ticket as ( P ). Then, under normal circumstances, the price for 5 tickets would be: [ 5P ] 2. **Determine the campaign price for 5 tickets:** In the campaign, 5 tickets can be purchased for the price of 3 tickets. Therefore, the price Mário paid for 5 tickets during the campaign is: [ 3P ] 3. **Calculate the savings:** The amount saved by Mário is the difference between the normal price and the campaign price: [ text{Savings} = 5P - 3P = 2P ] 4. **Determine the percentage saved:** The percentage savings is given by the ratio of the savings to the normal price, multiplied by 100: [ text{Percentage savings} = left( frac{2P}{5P} right) times 100% = left( frac{2}{5} right) times 100% ] 5. **Perform the calculation:** [ left( frac{2}{5} right) times 100% = 0.4 times 100% = 40% ] # Conclusion: The percentage savings for Mário is: [ boxed{40%} ]