Question

Cathy formed a three-digit number using the digits 1, 2, 3, 4. What is the probability that the sum of the digits of the three-digit number so formed is less than 8?

• A. $\frac{\text{1}}{\text{2}}$
• B. $\frac{\text{1}}{\text{3}}$
• C. $\frac{\text{2}}{\text{3}}$
• D. $\frac{\text{3}}{\text{4}}$

Answer 1

The correct option is A $\frac{\text{1}}{\text{2}}$

Let's write the total number of cases possible and their sum.
Three-digit number including digits 1, 2, 3:
123
132
213
231
312
321
In this case, the sum of the digits is always 6.

Three-digit number including the digits 1, 2, 4:
124
142
214
241
412
421
In this case, the sum of the digits is always 7.

Three-digit number including digits 1, 3, 4:
134
143
314
341
413
431
In this case, the sum of the digits is always 8.

Three-digit number including digits 2, 3, 4:
234
243
324
342
423
432
In this case, the sum of the digits is always 9.

Our favorable cases are 12 (cases include the digits 1, 2, 3 and 1, 2, 4) as only in these cases, the sum is either 6 or 7, which is less than 8.

Total number of cases = 24

Hence,
Probability = $\frac{\text{Favorable cases}}{\text{Total cases}}$

Probability = $\frac{\text{12}}{\text{24}}$ = $\frac{\text{1}}{\text{2}}$