The question doesn't mention unique words or repetition. How to know whether it is permutation or combination ?
Is it correct to say there are 7 letters in the word PROGRAM and R occurs 2 times, hence from 6 different letters there are 6C4 x 4C4 ways of arranging 4 letters from the word PROGRAM ?
I'm assuming the question is how many $4$ letter words can be formed from the letters in the word PROGRAM. All such words can be categorized as either containing no R's, containing $1$ R or containing $2$ R's.
Containing no R's : In how many ways can you select $4$ letters from P, O, G, A, M and arrange $4$ different letters?
Containing $1$ R : In how many ways can you select $3$ letters from P, O, G, A, M and arrange $4$ different letters?
Containing $2$ R's : In how many ways can you select $2$ letters from P, O, G, A, M and arrange $4$ letters out of which $2$ are the same?