Answer:
x² - 4x + 3
General Formulas and Concepts:
<u>Alg I</u>
Step-by-step explanation:
<u>Step 1: Define</u>
Roots: x = 1, 3
Leading Coefficient: 1
<u>Step 2: Write linear roots</u>
(x - 1)(x - 3)
<u>Step 3: Expand by FOIL</u>
x² - 3x - x + 3
<u>Step 4: Combine like terms</u>
x² - 4x + 3
How many ounces of lemonade was there in total? Once you get that divde by .25
1507 are the different ways can 5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players
<em><u>Solution:</u></em>
Given that,
5 baseball players and 4 basketball players be selected from 12 baseball players and 13 basketball players
This is a combination problem
Combinations are a way to calculate the total outcomes of an event where order of the outcomes does not matter
<em><u>The formula is given as:</u></em>
![n C_{r}=\frac{n !}{r !(n-r) !}](https://tex.z-dn.net/?f=n%20C_%7Br%7D%3D%5Cfrac%7Bn%20%21%7D%7Br%20%21%28n-r%29%20%21%7D)
Where n represents the total number of items, and r represents the number of items being chosen at a time
<em><u>Let us first calculate 5 baseball players from 12 baseball players</u></em>
Here, n = 12 and r = 5
![\begin{array}{l}{12 C_{5}=\frac{12 !}{5 !(12-5) !}} \\\\{12 C_{5}=\frac{12 !}{5 ! \times 7 !}}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B12%20C_%7B5%7D%3D%5Cfrac%7B12%20%21%7D%7B5%20%21%2812-5%29%20%21%7D%7D%20%5C%5C%5C%5C%7B12%20C_%7B5%7D%3D%5Cfrac%7B12%20%21%7D%7B5%20%21%20%5Ctimes%207%20%21%7D%7D%5Cend%7Barray%7D)
<em><u>For a number n, the factorial of n can be written as:</u></em>
![n !=n \times(n-1) \times(n-2) \times \ldots . \times 2 \times 1](https://tex.z-dn.net/?f=n%20%21%3Dn%20%5Ctimes%28n-1%29%20%5Ctimes%28n-2%29%20%5Ctimes%20%5Cldots%20.%20%5Ctimes%202%20%5Ctimes%201)
Therefore,
![\begin{aligned}12 C_{5} &=\frac{12 \times 11 \times 10 \times \ldots \ldots \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\\\12 C_{5} &=\frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2} \\\\12 C_{5} &=792\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D12%20C_%7B5%7D%20%26%3D%5Cfrac%7B12%20%5Ctimes%2011%20%5Ctimes%2010%20%5Ctimes%20%5Cldots%20%5Cldots%20%5Ctimes%202%20%5Ctimes%201%7D%7B5%20%5Ctimes%204%20%5Ctimes%203%20%5Ctimes%202%20%5Ctimes%201%20%5Ctimes%207%20%5Ctimes%206%20%5Ctimes%205%20%5Ctimes%204%20%5Ctimes%203%20%5Ctimes%202%20%5Ctimes%201%7D%20%5C%5C%5C%5C12%20C_%7B5%7D%20%26%3D%5Cfrac%7B12%20%5Ctimes%2011%20%5Ctimes%2010%20%5Ctimes%209%20%5Ctimes%208%7D%7B5%20%5Ctimes%204%20%5Ctimes%203%20%5Ctimes%202%7D%20%5C%5C%5C%5C12%20C_%7B5%7D%20%26%3D792%5Cend%7Baligned%7D)
<em><u>Similarly, 4 basketball players be selected 13 basketball players</u></em>
n = 13 and r = 4
Similarly we get,
![\begin{aligned}&13 C_{4}=\frac{13 !}{4 !(13-4) !}\\\\&13 C_{4}=\frac{13 !}{4 ! \times 9 !}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%2613%20C_%7B4%7D%3D%5Cfrac%7B13%20%21%7D%7B4%20%21%2813-4%29%20%21%7D%5C%5C%5C%5C%2613%20C_%7B4%7D%3D%5Cfrac%7B13%20%21%7D%7B4%20%21%20%5Ctimes%209%20%21%7D%5Cend%7Baligned%7D)
![13C_4 = 715](https://tex.z-dn.net/?f=13C_4%20%3D%20715)
<em><u>Thus total number of ways are:</u></em>
![12C_5 + 13C_4 = 792 + 715 = 1507](https://tex.z-dn.net/?f=12C_5%20%2B%2013C_4%20%3D%20792%20%2B%20715%20%3D%201507)
Thus there are 1507 different ways