How many numbers are there between 100 and 1000 such that atleast one of the their digits in 7?

In how many ways can the letters of the word PERMUTATIONS be arranged if the words start with P and end with S.

Determine n if ²ⁿC₃ : ⁿC₂ = 12 : 1

2n < (n + 2)! for all natural number n.

If the letters of the word APPLE are permuted in all possible ways and the strings then formed are arranged in the dictionary order show that the rank of the word APPLE is 12.