Question

Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$ and arranged in increasing order. The probability that the resulting sequence is an A.P. is

• A. $\frac{1}{2}$
• B. $\frac{3}{10}$
• C. $\frac{1}{5}$
• D. $\frac{1}{10}$
• E. $\frac{2}{5}$

Hint:

Combinatorics Tip: Whenever elements of a subset must be "arranged in increasing order," you only need to use Combinations ($C$), not Permutations ($P$). Any uniquely chosen subset can only be sorted in strictly increasing order in exactly one way!

Answer 1

Answer: (E)

Concept:
The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes. Since we are choosing exactly 3 distinct numbers from a set of 5, the total number of combinations is given by ${}^nC_r$. An Arithmetic Progression (A.P.) requires the difference between consecutive terms to be constant.

Step 1: Calculate the total number of possible combinations.
Choosing 3 numbers out of 5 corresponds to combinations (since the chosen numbers are always sorted in exactly one increasing order): $$\text{Total Outcomes} = {}^5C_3 = \frac{5!}{3!2!} = \frac{5 \times 4}{2} = 10$$

Step 2: Identify sequences with a common difference of 1.
We manually list the possible 3-term Arithmetic Progressions with a common difference $d = 1$: $$(1, 2, 3)$$ $$(2, 3, 4)$$ $$(3, 4, 5)$$ This gives 3 favorable outcomes.

Step 3: Identify sequences with a common difference of 2.
Now, check for Arithmetic Progressions with a common difference $d = 2$: $$(1, 3, 5)$$ This gives 1 favorable outcome.

Step 4: Sum the favorable outcomes.
Add the valid sequences found in the previous steps: $$\text{Favorable Outcomes} = 3 + 1 = 4$$

Step 5: Calculate the final probability.
Divide the favorable outcomes by the total outcomes: $$P(\text{A.P.}) = \frac{4}{10}$$ Simplify the fraction: $$P(\text{A.P.}) = \frac{2}{5}$$ Hence the correct answer is (E) $\frac{2{5}$}.