Permutations and Combinations: common misconceptions

If your child says… “Two choices and two choices always make four by addition.”

Ask together

If each last digit allowed three first digits, would adding still count every number?

What’s true

Multiply because each allowed last digit comes with its own complete set of first-digit choices.

If your child says… “At most one head means exactly one head.”

Ask together

Does TT have more than one head?

What’s true

At most includes smaller counts, while at least includes larger counts within the available total.

If your child says… “Zero factorial must be zero.”

Ask together

How many different arrangements remain when no positions need filling?

What’s true

Zero factorial is one because leaving every position empty gives one empty arrangement.

If your child says… “Swapping the Es makes another arrangement.”

Ask together

Without their numbers, how could you tell the exchanged Es apart?

What’s true

The two Es look identical, so their exchange produces no new letter arrangement.

If your child says… “Both roots answer the counting problem.”

Ask together

Could negative one objects supply the three positions?

What’s true

An algebraic root must also satisfy the original limits on available and chosen objects.

If your child says… “The pair counts as one thing, so its inside order never matters.”

Ask together

Does swapping A and B change the row while keeping them together?

What’s true

Keeping objects together still allows their internal order to change.

If your child says… “Different group sizes can never give equal selection counts.”

Ask together

Does every chosen pair identify exactly one object left behind?

What’s true

From the same pool, equal selection counts require equal sizes or sizes totalling the pool.

If your child says… “Choosing the pair already counts every row.”

Ask together

Does putting the pencil first give the same row as putting the pen first?

What’s true

Each selected pair makes two rows because exchanging its objects changes their positions.