Here we go:

1.Chef believes that a subset R of \{1, 2, \cdots, n\} is ripe if for any distinct numbers a, b \in R we have |a-b| > 3. You will be given a positive integer n. Find the number of ripe subsets of \{1, 2, \cdots, n\} modulo 10^4.Assume 1 \le n \le 10^5.

2.In this problem you will be given a positive integer n and a positive integer m. Count the number of solutions to the equation k_1k_2\cdots k_m = n in integers modulo 10^9 + 7.

Assume 1 \le n \le 10^4 and 2 \leq m \le 100.

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3.You will be given a string S of lower case english letters.Chef wants to partition this string into substrings so that each part contains at least one vowel(i.e. \text{a, e, i, o, u}). Find the number of ways in which Chef can do this modulo 10^9 + 7. For example when S = chefu we have the following partitions possible: chefu, che|fu, chef|u and so the respective answer in this case is 3.You can also check that the answer for S = zkzz is 0 because there are no vowels. Assume that 0 < |S| < 10^4.

Hope you enjoy the problems… (let me know if there is some error in any of the problem explanations)