How many ways can a total inventory of 30 batteries be distributed among the eight different types? 10,295,472
A camera shop stocks eight different types of batteries, one of which is type a7b. assume there are at least 30
Definition permutation (order is important)
No repetition allowed: [tex]P(n,r) = \frac{n!}{(n-r)!}[/tex]
Repetition allowed: [tex]n^r[/tex]
Definition combination (order is not important):
No repetition: [tex]C(n,r) = \left[\begin{array}{ccc}n\\r\end{array}\right] = \frac{n!}{r!(n-r)!}[/tex]
Repetition allowed: [tex]C(n+r-1,r) = \left[\begin{array}{ccc}n+r-1\\r\\\end{array}\right] = \frac{(n+r-1)!}{r!(n-1)!}[/tex]
with [tex]n! = n*(n-1)*...*2*1[/tex]
The camera shop stocks 8 different types of batteries and there are at least 30 batteries of each kind.
a) We want to select r = 30 batteries from the [tex]n=8[/tex] kinds of batteries
[tex]r=30\\n=8[/tex]
The order of the batteries doesn't matter, thus we should use a combination. Moreover, repetition is allowed as we can choose multiple pastries of the same kind.
[tex]\left[\begin{array}{ccc}30+8-1\\30\\\end{array}\right] = \left[\begin{array}{ccc}37\\30\end{array}\right][/tex]
[tex]= \frac{37!}{30!(37-30)!} \\= \frac{37!}{30!7!} \\= \frac{37*36*35*34*33*32*31*30!}{30!*(7*6*5*4*3*2*1)}[/tex]
[tex]n!=n*(n-1)*...*2*1[/tex]
[tex]=\frac{37*36*35*34*33*32*31}{7*6*5*4*3*2*1} \\=10,295,472[/tex]
Assuming that we first select 4 A76 batteries, we need to determine on how many ways we can select the remaining [tex]r=30-4=26[/tex] batteries from the [tex]n=8[/tex] kinds of batteries
The order of the batteries doesn't matter, thus we should use a combination. Moreover, the repetition is allowed as we can choose multiple pastries of the same kind.
[tex]\left[\begin{array}{ccc}26+8-1\\26\\\end{array}\right] = \left[\begin{array}{ccc}33\\26\end{array}\right][/tex]
[tex]= \frac{33!}{26!(33-26)!} \\= \frac{33!}{26!7!} \\= \frac{33*32*31*30*29*28*27*26!}{26!*(7*6*5*4*3*2*1)}[/tex]
[tex]n!=n*(n-1)*...*2*1[/tex]
[tex]=\frac{33*32*31*30*29*28*27*26!}{26!*(7*6*5*4*3*2*1} \\=4,272,048[/tex]
Grade: 5
Subject: math
Chapter: camera shop stocks
Keywords: stocks, camera, shop, batteries, inventory
