How many three-digit numbers are there, with no digit repeated?
Question:

How many three-digit numbers are there, with no digit repeated?

Solution:

Total number of 3-digit numbers $=$ Number of arrangements of 10 numbers, taken 3 at a time $={ }^{10} P_{3}=\frac{10 !}{7 !}=10 \times 9 \times 8=720$

Total number of 3-digit numbers, having 0 at its hundred’s place $={ }^{9} \mathrm{P}_{2}=\frac{9 !}{7 !}=9 \times 8=72$

Total number of 3-digit numbers with distinct digits $={ }^{10} P_{3}-{ }^{9} P_{2}=720-72=648$

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