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How many 6-digit integers greater than 400,000 can be formed

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How many 6-digit integers greater than 400,000 can be formed [#permalink] New post 03 Dec 2019, 02:47
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How many 6-digit integers greater than 400,000 can be formed such that each of the digits 2, 3, 4, 5, 6 and 7 is used once in each 6-digit integer?


A. 240

B. 480

C. 720

D. 960

E. 1,440


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[Reveal] Spoiler: OA

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Re: How many 6-digit integers greater than 400,000 can be formed [#permalink] New post 04 Dec 2019, 10:44
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How many 6-digit integers greater than 400,000 can be formed such that each of the digits 2, 3, 4, 5, 6 and 7 is used once in each 6-digit integer?


The question asks for a 6-digit number greater than 400,000.

The numbers that we can be used for the hundred-thousands digit is either 4, 5, 6 or 7. We have 4 choices.


4x?x?x?x?x?

For the ten-thousand digit, we can fit them with any number. Meaning, if we use 4 for the hundred-thousand, we can use the other 5 digits - 2, 3, 5, 6, 7. This also applies if we choose another number, let's say 2. We can then choose from 3, 4, 5, 6, 7. Again 5 digits, meaning 5 choices.

4x5x?x?x?x?

Use the same strategy for filling in the rest of the numbers and your equation looks like so.

4x5x4x3x2x1=?
[Reveal] Spoiler:

The answer is 480.

B.
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Re: How many 6-digit integers greater than 400,000 can be formed [#permalink] New post 05 Dec 2019, 02:13
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Lets tackle this in 2 cases.

Case 1: we have to make a 6 digit number and it will be greater than 400,000 if the first digit is among 5,6,7. This implies there are 3 ways to fill the first digit, 5 number are left. 2nd digit can be filled in 5 ways, 3rd digit in 4 ways and so on.
3 ways x 5 ways x 4 ways x 3 ways x 2 ways x 1 way = 360 ways.

Case 2: if the first digit is 4, it is filled in 1 way. The next digits can be filled just like we did in above case where every number will be greater than 400,000.
1 way x 5 ways x 4 ways x 3 ways x 2 ways x 1 way = 120 ways.

Summing them up, we have 480 ways.
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Re: How many 6-digit integers greater than 400,000 can be formed [#permalink] New post 07 Dec 2019, 10:37
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Re: How many 6-digit integers greater than 400,000 can be formed [#permalink] New post 07 Dec 2019, 10:38
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Re: How many 6-digit integers greater than 400,000 can be formed   [#permalink] 07 Dec 2019, 10:38
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