Brothers and sisters share the same two parents but are not identical (unless they are identical twins). The reason is in meiosis: two steps shuffle the chromosomes, so every gamete gets a different mix.
Key idea
Crossing over swaps matching pieces between homologous chromosomes in prophase I. Independent assortment lines each pair up on its own in metaphase I: with $n$ pairs there are $2^n$ ways to sort the chromosomes. Together they make every gamete unique.
Why it works
Independent assortment is like flipping a coin for each pair. With 2 pairs, each gamete can get either chromosome of pair 1 and either of pair 2: $2 \times 2 = 2^2 = 4$ combinations. With 3 pairs, $2^3 = 8$. Humans have 23 pairs, so $2^{23} = 8{,}388{,}608$ combinations, before crossing over adds even more. Crossing over happens when homologs pair up tightly: a non-sister chromatid of each homolog breaks at the same point and swaps the pieces, making chromosomes with new combinations of alleles.
Worked example: Count the combinations
A fruit fly has $2n = 8$. How many different combinations of chromosomes can its gametes have from independent assortment alone?
- The number of pairs is $n = 8 \div 2 = 4$.
- Each pair sorts two ways: $2^4 = 16$.
Answer: $16$
Worked example: Two gametes combine
For a species with $n = 3$, how many chromosome combinations are possible when one gamete from each of two parents combines (independent assortment only)?
- Each parent can make $2^3 = 8$ kinds of gamete.
- Any of one parent's 8 can pair with any of the other's 8: $8 \times 8 = 64$.
Answer: $64$
Common mistake
Thinking crossing over happens between sister chromatids, or that each pair's orientation affects the others. Sister chromatids are identical, so swapping pieces between them would change nothing; crossing over is between homologs (non-sister chromatids). And each pair lines up independently, like separate coin flips, which is why the count is $2^n$.