# The number of four digit numbers that can be formed from digits 0,1,2,3,4,5,6,7 if each number contains digit 1 is [closed]

What is the number of four digit numbers that can be formed from the digits $0,1,2,3,4,5,6,7$ so that each number contains digit $1$?

The answer is $750$ but I am not able to find it.

## closed as off-topic by uniquesolution, Rohan, Dave, hardmath, Ove AhlmanJan 18 '18 at 16:33

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• Please complete your sentence, you seem to have cut off there... – VortexYT Jan 18 '18 at 13:07
• Are the digits necessarily different? – user_194421 Jan 18 '18 at 13:57
• Yes , u cannot use one digit more than once – user163054 Jan 18 '18 at 14:06
• There are 4 choices for where the $1$ is located, and of the 3 other digits, the leftmost has 7 choices, the 2nd leftmost has 6 remaining choices, and the rightmost has 5 remaining choices. – Paul Sinclair Jan 18 '18 at 16:31

Since we are looking for $4$-digit numbers, the first (thousands) digit can't be $0$.

Let's split cases:

## Case 1: first digit is $1$.

In this case, we pick $3$ digits in order out of the remaining $7$, there are $7\cdot6\cdot5=210$ possibilities.

## Case 2: first digit is not $1$.

There are $6$ ways to choose the first digit ($2,3,...,7$), $3$ ways to place the digit $1$ (hundreds, tens, ones), and $6\cdot5$ (pick $2$ in order from the remaining $6$ digits) ways to choose the other digits. So there are $6\cdot3\cdot6\cdot5=540$ possibilities in this case.

## Total

In total, there are $210+540=750$ possibilities to choose the number.

• The answer is 480, I already mentioned that in question. – user163054 Jan 18 '18 at 15:09
• @user163054 If it's so, how do you know? – user_194421 Jan 18 '18 at 15:52
• I only have the answer, not complete solution – user163054 Jan 18 '18 at 15:58
• @user163054 Unless there are additional conditions that you did not mention, the correct answer is $750$. – N. F. Taussig Jan 18 '18 at 15:59
• There isn't anything additional. And this question appeared in a test and many students got the answer. – user163054 Jan 18 '18 at 16:01