How many ordered sequences are possible that contain seven objects chosen from fifteen?
ordered sequences

To determine how many **ordered sequences** are possible that contain **seven objects chosen from fifteen**, we are dealing with a **permutation problem** because the order of the objects matters.

The formula for the number of ordered sequences (or permutations) of \( r \) objects chosen from \( n \) objects is given by:

\[
P(n, r) = \frac{n!}{(n - r)!}
\]

In this case:
- \( n = 15 \) (the total number of objects),
- \( r = 7 \) (the number of objects to be chosen).

So the number of ordered sequences is:

\[
P(15, 7) = \frac{15!}{(15 - 7)!} = \frac{15!}{8!}
\]

Let's calculate that:

\[
15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8!
\]

The \( 8! \) cancels out, so we're left with:

\[
P(15, 7) = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9
\]

Calculating step-by-step:

\[
15 \times 14 = 210
\]
\[
210 \times 13 = 2730
\]
\[
2730 \times 12 = 32760
\]
\[
32760 \times 11 = 360360
\]
\[
360360 \times 10 = 3603600
\]
\[
3603600 \times 9 = 32432400
\]

Thus, the number of possible ordered sequences is:

\[
P(15, 7) = 32,432,400
\]

So, **32,432,400 ordered sequences** are possible.

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How many unordered selections are possible with the same parameters (combinations)?
2. What if the objects could repeat in the sequence (with repetition allowed)?
3. How many sequences would there be if only 5 objects were chosen from 15?
4. How does the formula change if there were no restrictions on how many times each object could be chosen?
5. What would the answer be if you were choosing all 15 objects instead of 7?

**Tip:** For large factorials, simplifying the cancellation of terms before direct multiplication often makes the calculation much easier.

How many ordered sequences are possible that contain seven objects chosen from fifteen?

Learn how to calculate the number of ordered sequences with seven objects chosen from fifteen using the permutation formula. This combinatorics problem involves factorial calculations and is suitable for students in Grades 10-12.

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