a) repetitions are allowed; b) repetitions not allowed
My solution: a) There are 3 cases:
- $3$-digit numbers;
- $2$-digit numbers;
- one digit numbers.
Case 1: $3\times 5\times5=75$
Case 2: $5\times5=25$
Case 3: $5 \times$ Sum of three cases $= 105$. (ans. key says $2253$?)
b) no repetitions:
Case 1: $3\times 4\times 3=36$
Case 2: $5\times 4=20$
Case 3: $5 \times$ Sum of cases $= 61$ (ans. key says $195$)
My answers are way different. Not sure what I'm missing. Help would be appreciated.
Your working out is correct for the two cases. I'm not sure exactly what the problem is -- perhaps you misread/mistranscribed the question? I'd recommend checking against just to make sure.