There are 4 suits so the chances of picking a spade initially are 1/4. There are 13 of each suit in a pack.
No spades: 39 other cards so if the first card is not a spade the probability is 39/52=3/4. Now there are 51 cards left, 13 of which are spades. The probability of no spade next is 38/51, then for the third card 37/50. The combined probability for no spades is 703/1700.
One spade: if a spade is drawn first the chances are 1/4. The next card is not a spade, chances are 39/51, and the third 38/50. Overall: 247/1700. Suppose the spade was drawn second: overall: 3/4 * 13/51 * 38/50=247/1700, and the same probability applies if the spade was third. So we have 3*247/1700=741/1700.
Two spades: this is similar to the non-spade being drawn in 1st, 2nd or 3rd position. We multiply one probability by 3. First two are spades: 1/4 * 12/51 * 39/50=39/850. Multiply by 3: 117/850.
Three spades: 1/4 * 12/51 * 11/50=11/850.
So we have:
#spades Probability
0 703/1700
1 741/1700
2 117/850
3 11/850
TOTAL: 1