Bob's Gift Shop sold 750 cards for Mother's Day. One salesman, Scarlett, sold 20% of the cards sold for Mother's Day. How many c
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Considering the definition of permutation, in 95,040 different ways he can arrange homes on the lots.
<h3>Definition of Permutation</h3>Permutation is placing elements in different positions. So, permutations of m elements in n positions are called the different ways in which the m elements can be arranged occupying only the n positions.
In other words, permutations are ways of grouping elements of a set in which:
- take all the elements of a set.
- the elements of the set are not repeated.
- order matters.
To obtain the total of ways in which m elements can be placed in n positions, the following expression is used:
where "!" indicates the factorial of a positive integer, which is defined as the product of all natural numbers before or equal to it.
<h3>This case</h3>In this case, you know:
- A contractor builds homes of 12 different models.
- Presently the constractor has 5 lots to build on.
To obtain the total of ways in which he can arrange homes on these lots, you use the permutation. This is, 12 elements o models can be arranged occupying only the 5 positions:
Solving:
12P5= 479,001,600 ÷ 5,040
<u><em>12P5=95,040</em></u>
Finally, in 95,040 different ways he can arrange homes on the lots.
Learn more about permutation:
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Let 
. Then the definite integral can be split up at 
so that
Now take the derivative. By the fundamental theorem of calculus, you have
![\displaystyle\frac{\mathrm dF}{\mathrm dx}=\frac{\mathrm d}{\mathrm dx}\left[\int_{t=c}^{t=5x}\frac{\mathrm dt}t-\int_{t=c}^{t=2x}\frac{\mathrm dt}t\right]](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B%5Cmathrm%20dF%7D%7B%5Cmathrm%20dx%7D%3D%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cint_%7Bt%3Dc%7D%5E%7Bt%3D5x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt-%5Cint_%7Bt%3Dc%7D%5E%7Bt%3D2x%7D%5Cfrac%7B%5Cmathrm%20dt%7Dt%5Cright%5D)
![=\displaystyle\frac1{5x}\dfrac{\mathrm d}{\mathrm dx}[5x]-\frac1{2x}\dfrac{\mathrm d}{\mathrm dx}[2x]](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle%5Cfrac1%7B5x%7D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B5x%5D-%5Cfrac1%7B2x%7D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B2x%5D)
Then integrating with respect to
, we recover 
and find that
where
is an arbitrary constant.
Now take the derivative. By the fundamental theorem of calculus, you have
Then integrating with respect to
where