Math Problem Statement

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

Solution

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 rr objects chosen from nn objects is given by:

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

In this case:

  • n=15n = 15 (the total number of objects),
  • r=7r = 7 (the number of objects to be chosen).

So the number of ordered sequences is:

P(15,7)=15!(157)!=15!8!P(15, 7) = \frac{15!}{(15 - 7)!} = \frac{15!}{8!}

Let's calculate that:

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

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

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

Calculating step-by-step:

15×14=21015 \times 14 = 210 210×13=2730210 \times 13 = 2730 2730×12=327602730 \times 12 = 32760 32760×11=36036032760 \times 11 = 360360 360360×10=3603600360360 \times 10 = 3603600 3603600×9=324324003603600 \times 9 = 32432400

Thus, the number of possible ordered sequences is:

P(15,7)=32,432,400P(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.

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Math Problem Analysis

Mathematical Concepts

Permutations
Combinatorics
Factorial

Formulas

Permutation formula: P(n, r) = n! / (n - r)!

Theorems

Permutation Theorem

Suitable Grade Level

Grades 10-12