Questions: How many different ID cards can be made if there are six digits on a card and no digit can be used more than once? What if digits can be repeated? Part 1 of 2 There are different ID cards that can be made if no digit can be used more than once. Part 2 of 2 There are different ID cards that can be made if digits can be repeated.

How many different ID cards can be made if there are six digits on a card and no digit can be used more than once? What if digits can be repeated?

Part 1 of 2

There are different ID cards that can be made if no digit can be used more than once.

Part 2 of 2

There are different ID cards that can be made if digits can be repeated.

Transcript text: How many different ID cards can be made if there are six digits on a card and no digit can be used more than once? What if digits can be repeated? Part 1 of 2 There are $\square$ different ID cards that can be made if no digit can be used more than once. Part 2 of 2 There are $\square$ different ID cards that can be made if digits can be repeated.

Solution

Solution Steps

Step 1: Understand the Problem

We need to calculate the number of different ID cards that can be made with 6 digits, where no digit is repeated.

Step 2: Apply the Permutation Formula

Given that we're using a decimal system (0-9), we have 10 unique digits. The formula to calculate the number of different ID cards without repetition is $P(n, k) = \frac{n!}{(n-k)!}$, where $n=10$ and $k= 6 $.

Step 3: Perform the Calculation

Substituting the values into the formula, we get $P(10, 6 ) = \frac{10!}{(10- 6 )!} = 151200 $.

Final Answer:

The number of different ID cards that can be made with 6 digits, without repeating any digit, is 151200 .

Step 1: Understand the Problem

We need to calculate the number of different ID cards that can be made with 6 digits, where digits can be repeated.

Step 2: Apply the Formula for Repetition

Each digit on the ID card can be any of the 10 digits (0-9), and this choice is independent for each of the 6 positions on the card. The formula to calculate the number of different ID cards with repetition allowed is $10^n$.

Step 3: Perform the Calculation

Substituting the value of $n= 6 $ into the formula, we get $10^ 6 = 1000000 $.

Final Answer:

The number of different ID cards that can be made with 6 digits, allowing repetition of digits, is 1000000 .

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