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# Combinations

## The number of possible groupings when order does not matter.

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Blue Eyes or Brown Eyes?

Credit: Angela Mueller
Source: http://www.flickr.com/photos/22103696@N07/4280548041/

Have you ever wondered why a person can have blue eyes when everyone else in their family has brown eyes? Or why some friends have curly hair when their parents both have straight hair? Understanding heredity is an easy way to begin to understand genetics or which traits are inherited by offspring from their parents. This information is used every day in fields like agriculture, medicine, and cancer research. You can even use the physical traits of offspring to determine the genetic make-up of parents and vice versa.

#### Why It Matters

The genes you carry in your DNA determine your biological traits, and some of those, like eye color, are physical characteristics. Your DNA is composed of 46 chromosomes—you inherited 23 from your mother and 23 from your father and these chromosomes pair up to make your DNA. A gene is a section of DNA. Your genes are comprised of pairs of information (remember, one half from your father and one half from your mother), and these pairs are called alleles and determine the appearance of physical characteristics.

In the case of eye color, we can simplify this idea a bit. The allele for brown eyes is B (we make it a capital letter because it is the dominant gene, or the one that will show up if you have it) and the allele for blue eyes is b (we make it a lowercase letter because a person only has blue eyes if they have two b alleles, one from each parent, meaning the gene is recessive).

Credit: Steven Terrebonne
Source: CK-12 Foundation

So, where does the math come in? If you have two parents that have the eye-color genes Bb and Bb and they have a baby, what is the probability that the baby has blue eyes? To figure this out, we need to look at all of the possible allele combinations that the baby could inherit from her parents: BB, Bb, bB, bb. We recognize this as a probability of combinations problem! In other words, what is the probability of choosing two b alleles (only the combination bb in the baby will result in blue eyes)? From each parent, we choose one allele and there are two possibilities:

2C12C1=2!(21)!1!2!(21)!1!=21211111=4\begin{align*}_2C_1\cdot _2C_1=\frac{2!}{(2-1)!\cdot 1!}\cdot \frac{2!}{(2-1)!\cdot 1!}=\frac{2 \cdot 1 \cdot 2 \cdot 1}{1 \cdot 1 \cdot 1 \cdot 1}=4\end{align*}

So, there are 4 total possible combinations and only one of those combinations is bb. Therefore the probability that the baby has blue eyes is 14=25%\begin{align*}\frac{1}{4}=25\%\end{align*}.

See for yourself using this animated tutorial:
http://learn.genetics.utah.edu/content/begin/tour/index.html

#### Explore More

If having dimples is the dominant trait and only due to one gene and neither of Jeanne's parents have dimples, what is the probability that Jeanne has dimples? (Hint: Use D as the allele for dimples and d as the allele for no dimples.)

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