Question

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $a/b$, where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$, then $a + b$ is equal to _______

Answer 1

Correct Answer: 944

Step 1: Understanding the Concept:
To find the probability, we must determine the total number of ways to pick 3 distinct dates and the number of ways those 3 dates form an Arithmetic Progression (A.P.). A critical property of an A.P. triplet $(x, y, z)$ is that $x + z = 2y$, which implies $x$ and $z$ must share the same parity (both even or both odd).

Step 2: Key Formula or Approach:
Total ways to select 3 items from $n$ is $\binom{n}{3}$.
An A.P. of length 3 is uniquely determined by its endpoints $x$ and $z$ as long as they have the same parity. The number of such pairs is $\binom{n_{even}}{2} + \binom{n_{odd}}{2}$.

Step 3: Detailed Explanation:
Total number of ways to select 3 dates out of 31:
$n(S) = \binom{31}{3} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1} = 31 \times 5 \times 29 = 4495$.
Let the 3 dates in an increasing A.P. be $x, y, z$ where $x<y<z$.
Because they are in an A.P., the common difference is $d = y - x = z - y$.
Thus, $x + z = 2y$.
Since $2y$ is always an even integer, the sum $x + z$ must be even. This is only possible if both $x$ and $z$ are even or both are odd.
Once any valid pair $(x, z)$ of the same parity is chosen, $y$ is automatically fixed as their midpoint.
In 31 days, there are:
Odd dates: $1, 3, 5, \dots, 31$ (Total 16 odd dates)
Even dates: $2, 4, 6, \dots, 30$ (Total 15 even dates)
Number of ways to choose 2 odd dates: $\binom{16}{2} = \frac{16 \times 15}{2} = 120$.
Number of ways to choose 2 even dates: $\binom{15}{2} = \frac{15 \times 14}{2} = 105$.
Total number of favorable ways (A.P. combinations) = $120 + 105 = 225$.
Probability $P = \frac{225}{4495}$.
Let's simplify the fraction:
Divide by 5: $\frac{45}{899}$.
Since $899 = 30^2 - 1 = 29 \times 31$, and $45 = 9 \times 5$, they share no common factors.
Thus, $\gcd(45, 899) = 1$.
Here, $a = 45$ and $b = 899$.
Calculate $a + b$:
$a + b = 45 + 899 = 944$.

Step 4: Final Answer:
The value of $a + b$ is 944.