There are a total of 2^{5} or 32 outcomes (e.g. if T denotes a tail and H denotes a head, then the sample space will have 32 possibilities: TTTTT, HTTTT, THTTT, ..., THHHH, HHHHH)

0 heads —> ^{5}C_{0} —> 1 outcome

Probability of 0 heads = ^{1}⁄_{32} = 0.03125

1 heads —> ^{5}C_{1} —> 5 outcomes

Probability of 1 heads = ^{5}⁄_{32} = 0.15625

2 heads —> ^{5}C_{2} —> 10 outcomes

Probability of 2 heads = ^{10}⁄_{32} = 0.3125

3 heads —> ^{5}C_{3} —> 10 outcomes

Probability of 3 heads = ^{10}⁄_{32} = 0.3125

4 heads —> ^{5}C_{4} —> 5 outcomes

Probability of 4 heads = ^{5}⁄_{32} = 0.15625

5 heads —> ^{5}C_{5} —> 1 outcome

Probability of 5 heads = ^{1}⁄_{32} = 0.03125

(Check)
Sum of probabilities of all outcomes = 0.03125 + 0.15625 + 0.3125 + 0.3125 + 0.15625 + 0.03125 = 1

Probability of getting 1 to 3 heads = (Probability of 1 head) + (Probability of 2 heads) + (Probability of 3 heads) = 0.15625 + 0.3125 + 0.3125 = 0.78125