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.
2026-02-22 19:36:43.1771789003
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
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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.