How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

Question

Accepted Answer

> [!TLDR]
> We need to count 7-digit numbers (since > 1,000,000) formed from the digits 1, 2, 0, 2, 4, 2, 4, with the restriction that the first digit cannot be 0. The digits have repetitions: three 2’s, two 4’s, one 1, one 0. The total count is **360**.

The question asks for numbers greater than 1,000,000. That means the number must have at least 7 digits. Since we are given exactly 7 digits (1, 2, 0, 2, 4, 2, 4), every number we form will be a 7-digit number. But a 7-digit number cannot start with 0 — otherwise it would effectively be a 6-digit number. So the first digit must be chosen from the non-zero digits available.

This is a classic permutations-with-repetition problem. When some items are identical, the number of distinct arrangements of $n$ items where there are $p$ of one kind, $q$ of another, etc., is:

> [!FORMULA]
> $$ \frac{n!}{p! \, q! \, \cdots} $$

Here, $n = 7$, with three 2’s, two 4’s, one 1, and one 0. So the total arrangements without any restriction would be:

$$ \frac{7!}{3! \, 2! \, 1! \, 1!} = \frac{5040}{6 	imes 2} = \frac{5040}{12} = 420. $$

But this includes arrangements where the first digit is 0, which are invalid. So we need to subtract those.

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**Step-by-step reasoning:**

1. **Total unrestricted arrangements**  
   As above, total distinct 7-digit sequences (allowing leading zero) = $420$.

2. **Count arrangements where the first digit is 0**  
   If the first digit is fixed as 0, we have 6 remaining positions to fill with the remaining digits: three 2’s, two 4’s, and one 1.  
   Number of distinct arrangements of these 6 digits:

$$ \frac{6!}{3! \, 2! \, 1!} = \frac{720}{6 	imes 2} = \frac{720}{12} = 60. $$

3. **Subtract invalid cases**  
   Valid numbers = total − those starting with 0 = $420 - 60 = 360$. …

Q.How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

Concept understanding — Permutations Without Repetition

Permutations Without Repetition – The Idea of Arranging Things

Why "Without Repetition"?

The Counting Logic – Why Multiply?

What If You Only Arrange Some of Them?

Key Points to Remember

Common Mistake to Avoid …

Q.How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

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