Maths Q9 – JEE-MAIN 2025

Home Maths Permutations and CombinationsQuestion #324

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to :

Clue 1 of 3: Conceptual Principle

This problem involves arranging a set of objects where some objects are identical.

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● Video Walkthrough

Step 1: Determine the count of each digit✦ Active

The sequence has ten terms. It contains exactly five 1s and exactly three 2s. The remaining terms must be 0s. The number of 0s is $10 - 5 - 3 = 2$ .

n = 10 (total terms)

n_{1} = 5 (number of 1s)

n_{2} = 3 (number of 2s)

n_{0} = 10 - 5 - 3 = 2 (number of 0s)

Step 2: Apply the multinomial coefficient formula○ Expand

The number of distinct sequences (permutations) of $n$ objects where there are $n_{1}$ identical objects of type 1, $n_{2}$ identical objects of type 2, and $n_{0}$ identical objects of type 0 is given by the formula:

Number of sequences = \frac{n!}{n_{1}! n_{2}! n_{0}!}

💡 Teacher's Secret Hint

Ensure all counts sum up to the total number of terms.

Step 3: Calculate the final value○ Expand

Substitute the values into the formula:

Number of sequences = \frac{10!}{5! \cdot 3! \cdot 2!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5! \times (3 \times 2 \times 1) \times (2 \times 1)}

= \frac{10 \times 9 \times 8 \times 7 \times 6}{6 \times 2}

= 10 \times 9 \times 4 \times 7

= 360 \times 7

= 2520

💡 Teacher's Secret Hint

Simplify the factorials carefully to avoid calculation errors.

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Maths Question 9 – JEE-MAIN 2025

Clue 1 of 3: Conceptual Principle

🧠 Full Solution Path