David Lewis

2021-12-17

A camera shop stocks eight different types of batteries, one of which is type A7b.Assume there are at least 30 batteries.

of each type.a. How many ways can a total inventory of 30 batteries be distributed among the eight different types? b.

How many way can a total inventory of 30 batteries be distributed among the eight different types of the inventory must

include at least four A76 batteries?c. How many ways can a total inventory of 30 batteries be distributed among the eight

different types of the inventory includes at most three A7b batteries

of each type.a. How many ways can a total inventory of 30 batteries be distributed among the eight different types? b.

How many way can a total inventory of 30 batteries be distributed among the eight different types of the inventory must

include at least four A76 batteries?c. How many ways can a total inventory of 30 batteries be distributed among the eight

different types of the inventory includes at most three A7b batteries

Juan Spiller

Beginner2021-12-18Added 38 answers

Denitions

Definition permulatin (oder is important)

No repetition allowed:$P(n,r)=\frac{n!}{(n-r)!}$

Repetition allowed:$n}^{r$

Definition combination(order is not important):

No repetition allowed:$$(\begin{array}{c}n\\ r\end{array})=\frac{n!}{r!(n-r)!}$$

Repetition allowed:$$C(n+r-1,r)=(\begin{array}{c}n+r-1\\ r\end{array})=\frac{(n+1-1)!}{r!(n-1)!}$$

with$n\ne n\cdot (n-1)\cdot \cdots c\cdot 2\cdot 1$

Solution

The camera shop stokcs8 different types of batteries and there are at least 30 batteries of each kind

(a) We want to select r=30 batteries frrom the n=8 kinds of batteries,

$r=30$

$n=8$

The order of the batteries doesnt

Definition permulatin (oder is important)

No repetition allowed:

Repetition allowed:

Definition combination(order is not important):

No repetition allowed:

Repetition allowed:

with

Solution

The camera shop stokcs8 different types of batteries and there are at least 30 batteries of each kind

(a) We want to select r=30 batteries frrom the n=8 kinds of batteries,

The order of the batteries doesnt

stomachdm

Beginner2021-12-19Added 33 answers

a) No of ways can a total inventory of

30 batteries be distributed among the

eight different types

$$=\text{}(\begin{array}{c}30+8-1\\ 30\end{array}\text{})=\text{}(\begin{array}{c}37\\ 30\end{array}\text{})$$

$=\frac{37!}{30!7!}=\frac{37\times 36\times 35\times 34\times 33\times 32\times 31}{7\times 6\times 5\times 4\times 3\times 2\times 1}$

$10295472$

b) The number of ways can a total inventory

of 30 batteries be distributed among 8

different types if the inventory must include

at least 4 A76 batteries

$$=\sum _{i=1}^{3}(\text{}\begin{array}{c}(30-k)+7-1\\ 30-k\end{array}\text{})=(\text{}\begin{array}{c}36\\ 30\end{array}\text{})+(\text{}\begin{array}{c}35\\ 29\end{array}\text{})(\text{}\begin{array}{c}34\\ 28\end{array}\text{})+(\text{}\begin{array}{c}33\\ 27\end{array}\text{})$$

$=1947792+1623160+1344904+1107568$

$=6,023,424$

30 batteries be distributed among the

eight different types

b) The number of ways can a total inventory

of 30 batteries be distributed among 8

different types if the inventory must include

at least 4 A76 batteries

nick1337

Expert2021-12-27Added 777 answers

(a)

To calculate:

The number of ways a batch of 30 batteries can be distributed among eight

categories.

Formula used:

The namber of ways to select r number of elements from a set of n elements is

given by:

Calculation

It is nedded to select an inventory of 30 individuals from eight diffeent types and

it is possible to select all units from one types or miss one or few types in the

selection.

Let n=8, r=30.

(b)

To find :

Thenumber of possibilities to select inventore of 30 batteries representing all eught

categories including at least 4 batteries form category A76.

Calculation

In part(a) we have obtained the number is shows that possible ways to select 30

batteries. By subtracting the number of ways to select 30 batteries includung ar

most 3 units of A76 from this total count, the number of ways that can be select a

batch of 30 batteries with at least 4 batteries of A76 can be found.

Now substitute n=7, r=29.

[

Then, the selection of 2 units from A776 is substituting n=7, r=28.

Then, the selection of 3 units from A76 is substitutiong n=7, r=27.

If there is no A76 batteries then n=7, r=30.

Thus, the total number of possible ways to selecting 30 batteries,

(c)

To find:

The number of possible ways to select a batch of 30 batteries including at most 3

batteries from the type AT6

Calculation:

The number of possible ways to select a batch of 30 batteries includung at most 3

batteries from the type A76 can be calculated by subtract nymber of possible ways

to selecting 30 batteries including at least 4 of A76 from total possibility.

That is

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If some function like $f$ depends on just one variable like $x$, we denote its derivative with respect to the variable by:

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Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:

$\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}f({x}_{1},\cdots ,{x}_{i},\cdots ,{x}_{n})$

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1) $F(x,y)$ 𝑎𝑛𝑑 $y=f(x)$ so his means that the function $F$ is a function of one variable which is $x$

2) while we were computing 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 we treated $y$ and $x$ as two independent variables although that $y$ changes as $x$ changes but while doing the 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 w.r.t $x$ we treated $y$ and $x$ as two independent varaibles and considered $y$ as a constantLet $f:{\mathbb{R}}^{2}\to \mathbb{R}$ be defined as

$f(x,y)=\{\begin{array}{ll}({x}^{2}+{y}^{2})\mathrm{cos}\frac{1}{\sqrt{{x}^{2}+{y}^{2}}},& \text{for}(x,y)\ne (0,0)\\ 0,& \text{for}(x,y)=(0,0)\end{array}$

then check whether its differentiable and also whether its partial derivatives ie ${f}_{x},{f}_{y}$ are continuous at $(0,0)$. I dont know how to check the differentiability of a multivariable function as I am just beginning to learn it. For continuity of partial derivative I just checked for ${f}_{x}$ as function is symmetric in $y$ and $x$. So ${f}_{x}$ turns out to be

${f}_{x}(x,y)=2x\mathrm{cos}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)+\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\mathrm{sin}\left(\frac{1}{\sqrt{{x}^{2}+{y}^{2}}}\right)$

which is definitely not $0$ as $(x,y)\to (0,0)$. Same can be stated for ${f}_{y}$. But how to proceed with the first part?