Maths Q14 – AP-EAMCET 2026

Home Maths Permutations and CombinationsQuestion #307

Let A be a set of

n (\geq 3)

distinct elements. For

x, y, z \in A

, if

A^{3}

is the set of all ordered triplets

(x, y, z)

, then the number of triplets of

A^{3}

in which at least two among

x, y, z

are equal is

Clue 1 of 3: Complementary Counting

Often, when a problem asks for 'at least' a certain condition, it's easier to calculate the total number of possibilities and subtract the number of possibilities where the condition is NOT met (i.e., its complement).

🧠 Full Solution Path

Step 1: Identify the Total Number of Ordered Triplets✦ Active

Let $A$ be a set with $n$ distinct elements. We are considering ordered triplets $(x, y, z)$ where $x, y, z \in A$ . Since each position can be filled by any of the $n$ elements, the total number of such ordered triplets is:

Total triplets = n \times n \times n = n^{3}

💡 Teacher's Secret Hint

Remember that an 'ordered triplet' means the order of elements matters, and repetition is allowed unless stated otherwise. Here, $x, y, z$ can be the same.

Step 2: Identify the Complement Event○ Expand

The problem asks for the number of triplets in which 'at least two among $x, y, z$ are equal'. This condition is equivalent to (Total triplets) - (Triplets where all three elements are distinct). The complement event is when all three elements $x, y, z$ are distinct.

💡 Teacher's Secret Hint

Problems with 'at least' or 'at most' often simplify significantly using the complementary counting principle.

Step 3: Calculate Number of Triplets with All Distinct Elements○ Expand

For the elements $x, y, z$ to be distinct, we choose one element for $x$ , another different element for $y$ , and a third different element for $z$ . This is a permutation of $n$ elements taken 3 at a time:

Triplets with distinct elements = n P_{3} = n (n - 1) (n - 2)

Expanding this expression:

n (n - 1) (n - 2) = n (n^{2} - 3 n + 2) = n^{3} - 3 n^{2} + 2 n

💡 Teacher's Secret Hint

Ensure you correctly expand the product. A common error is miscalculating the coefficients.

Step 4: Calculate Number of Triplets with At Least Two Equal Elements○ Expand

Now, subtract the number of triplets with distinct elements from the total number of triplets:

Triplets with at least two equal = Total triplets - Triplets with distinct elements

= n^{3} - (n^{3} - 3 n^{2} + 2 n)

= n^{3} - n^{3} + 3 n^{2} - 2 n

= 3 n^{2} - 2 n

💡 Teacher's Secret Hint

Double-check the signs when removing the parentheses. A common mistake is $- (A - B) = - A - B$ instead of $- A + B$ .

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Maths Question 14 – AP-EAMCET 2026

Clue 1 of 3: Complementary Counting

🧠 Full Solution Path