Hello, a69356!

I will assume that the numberHow many six-digit numbers contains exactly 4 different digits?maybegin with zero.

First, select 4 digits from the available 10 digits.

. . There are: . choices.

Then there are two possible distributions of the digits.

. . .

Case (1): .

. . We have a Triple and three Singletons.

There are 4 choices for the Triple

Then the letters can be arranged in: ways.

There are: .

Case (2): .

. . We have two Pairs and two Singletons.

There are: choices for the two Pairs.

Then the letters can be arranged in: ways.

There are: .

Therefore, there are: . six-digit numbers

. . that contain four different digits.

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If leading zeros are not allowed, consider this fact:

. . One-tenth of the above numbers begin with zero.