Volume 2: The Logic of Creation

Workbook 19.1: The Manifold Wisdom

Directives for the Weaver:

1. Ask the Swap Question: If I swap two items, is the result different?
   - Yes = Permutation ($nPr$). (Order Matters).
   - No = Combination ($nCr$). (Order Doesn't Matter).
2. Factorial Check: $n!$ means $n \times (n-1) \times (n-2)...$ down to 1.
3. $0! = 1$: By definition, there is exactly one way to arrange "nothing."

Part I: Factorial Foundations

Calculate these total arrangements.

$4! = $

$4 \times 3 \times 2 \times 1 = 24$

$6! / 4! = $

$(6 \times 5 \times 4!) / 4! = 6 \times 5 = 30$

The Alphabet: How many ways can you arrange the letters in the word "FAITH"?

5 letters... 5! = ...

Part II: Identifying the Logic (P or C?)

Mark each scenario as either a Permutation (P) or a Combination (C).

Choosing a President, Vice President, and Secretary from a group of 10 people.

[ ] P   [ ] C   (Why? Does position matter?)

Choosing a "cleaning crew" of 3 people from a group of 10.

[ ] P   [ ] C   (Why?)

Creating a 4-digit PIN for your bank account.

[ ] P   [ ] C

Choosing 5 books from the library to take home for the week.

[ ] P   [ ] C
The Unity Check:

In the "cleaning crew" problem, if you chose {Peter, James, John}, would that be a different crew than {John, James, Peter}? Explain.

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Part III: Calculating the Choices

Use the formulas: $nPr = \frac{n!}{(n-r)!}$ and $nCr = \frac{n!}{r!(n-r)!}$.

The Race: 8 runners are in a race. How many different ways can the Gold, Silver, and Bronze medals be awarded?

$n=8, r=3$. Order matters (P).
$8P3 = 8 \times 7 \times 6 = ...$

The Missionary Team: A church has 10 members willing to go on a trip. They only have room for 4. How many different teams of 4 can be made?

$n=10, r=4$. Order doesn't matter (C).
$10C4 = (10 \times 9 \times 8 \times 7) / (4 \times 3 \times 2 \times 1) = ...$

Part IV: The Challenge (The Breastplate)

The Tribal Order

Moses needs to choose 3 tribes out of the 12 to lead the march.
1. If the order of the 3 tribes in the line matters, how many arrangements are there?
2. If he just needs a committee of 3 tribes to meet with him, how many combinations are there?

Permutations ($12P3$):
Combinations ($12C3$):
Notice how much smaller the committee number is!

Part V: Transmission (The Echad Extension)

Teacher Log: The Toy Box Shuffle

Objective: Explain the difference between order and group to a younger sibling.

The Activity: Use 3 toys (A, B, C).
1. Ask them to line them up for a "parade." How many ways? (6).
2. Ask them to just put 2 toys in a bag. How many ways? (3: AB, AC, BC).

The Lesson: "Parades have ranks (1st, 2nd, 3rd). Bags have friends (they are just together). In God's family, we are 'friends' in the bag first!"


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