Quick questions on Distinct arrangements of objects with repeats: the n! over r1! ... rk! formula and the binary two-type special case

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is separating identical items, using the complement?

Show answer

"Must be separated" works exactly as it does for distinct items, with one simplification: an identical pair has no internal order to track. To count arrangements where two identical items are apart, count the total, then subtract the arrangements where they are together (found by gluing them into one block), because "apart" is the complement of "together".

What is all patterns:

2^n

Show answer

Each of the

n

positions is filled independently with one of two symbols, so by the multiplication principle the number of patterns is

2 \times 2 \times \cdots \times 2 = 2^n

. For the eight lights,

2^8 = 256

What is exactly

r

of one kind: a combination?

Show answer

Fix how many are red, say

r

, and a pattern is determined entirely by which

r

of the

n

positions are red; the rest are green. That is a selection of positions,

^{n}C_{r}

. Equivalently it is the identical-elements formula for a word of

r

reds and

n - r

greens,

Have a question we have not covered?

This dot-point answer is short enough that we have not extracted many short questions yet. Read the full dot-point answer or ask Mo, our study assistant, in the chat for follow ups.

All Maths Extension 1Q&A pages