If for 3 <= r <= 30, (30C(30-r)) + 3(30C(31-r)) + 3(30C(32-r)) + (30C(33-r)) = mCr, then m equals:

The correct answer is Option 3. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q6 – JEE-MAIN 2026

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Home Maths Binomial TheoremQuestion #36

If for

3 \leq r \leq 30

(\binom{30}{30 - r}) + 3 (\binom{30}{31 - r}) + 3 (\binom{30}{32 - r}) + (\binom{30}{33 - r}) = (\binom{m}{r})

, then

m

equals:

Clue 1 of 3: Binomial Coefficient Properties

Recall the symmetry property of binomial coefficients, $(\binom{n}{k}) = (\binom{n}{n - k})$ , to simplify the terms.

🧠 Full Solution Path

Step 1: Apply Symmetry Property✦ Active

First, use the property $(\binom{n}{k}) = (\binom{n}{n - k})$ to rewrite the given terms. This simplifies the lower index of the binomial coefficients.

(\binom{30}{30 - r}) = (\binom{30}{r})

(\binom{30}{31 - r}) = (\binom{30}{30 - (31 - r)}) = (\binom{30}{r - 1})

(\binom{30}{32 - r}) = (\binom{30}{30 - (32 - r)}) = (\binom{30}{r - 2})

(\binom{30}{33 - r}) = (\binom{30}{30 - (33 - r)}) = (\binom{30}{r - 3})

The expression becomes:

(\binom{30}{r}) + 3 (\binom{30}{r - 1}) + 3 (\binom{30}{r - 2}) + (\binom{30}{r - 3})

Step 2: Apply Pascal's Identity Iteratively○ Expand

Rearrange the terms and apply Pascal's identity, $(\binom{n}{k}) + (\binom{n}{k - 1}) = (\binom{n + 1}{k})$ , multiple times. We can write $3 (\binom{30}{r - 1})$ as $(\binom{30}{r - 1}) + 2 (\binom{30}{r - 1})$ and $3 (\binom{30}{r - 2})$ as $2 (\binom{30}{r - 2}) + (\binom{30}{r - 2})$ .

((\binom{30}{r}) + (\binom{30}{r - 1})) + 2 ((\binom{30}{r - 1}) + (\binom{30}{r - 2})) + ((\binom{30}{r - 2}) + (\binom{30}{r - 3}))

Applying Pascal's identity to each pair:

(\binom{31}{r}) + 2 (\binom{31}{r - 1}) + (\binom{31}{r - 2})

Now, split $2 (\binom{31}{r - 1})$ into $(\binom{31}{r - 1}) + (\binom{31}{r - 1})$ and apply Pascal's identity again:

((\binom{31}{r}) + (\binom{31}{r - 1})) + ((\binom{31}{r - 1}) + (\binom{31}{r - 2}))

Applying Pascal's identity:

(\binom{32}{r}) + (\binom{32}{r - 1})

💡 Teacher's Secret Hint

Ensure the lower index for Pascal's identity is consistent. For $(\binom{n}{k}) + (\binom{n}{k - 1}) = (\binom{n + 1}{k})$ , the larger lower index is $k$ in the result.

Step 3: Final Application of Pascal's Identity and Determine m○ Expand

Apply Pascal's identity one last time to the simplified expression:

(\binom{32}{r}) + (\binom{32}{r - 1}) = (\binom{33}{r})

Given that the original expression equals $(\binom{m}{r})$ , we have:

(\binom{33}{r}) = (\binom{m}{r})

Comparing both sides, we find that $m = 33$ .

💡 Teacher's Secret Hint

The range $3 \leq r \leq 30$ ensures all binomial coefficients are well-defined and valid for the application of these identities.

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Maths Question 6 – JEE-MAIN 2026

Clue 1 of 3: Binomial Coefficient Properties

🧠 Full Solution Path