Combinatorics
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I greet you this day,
For the Classic ACT exam:
The ACT Mathematics test is a timed exam...60 questions in 60 minutes
This implies that you have to solve each question in one minute.
Each of the first 20 questions (less challenging) will typically take less than a minute a solve.
Each of the next 20 questions (medium challenging) may take about a minute to solve.
Each of the last 20 questions (more challenging) may take more than a minute to solve.
The goal is to maximize your time.
You use the time saved on the questions you solve in less than a minute to solve questions that will take more
than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.
Also: please note that unless specified otherwise, geometric figures are drawn to scale. So, you can figure out
the correct answer by eliminating the incorrect options.
Other suggestions are listed in the solutions/explanations as applicable.
These are the solutions to the ACT past questions on the topics in Combinatorics.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided
for some questions.
The link to the video solutions will be provided for you. Please
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Please NOTE: For applicable questions involving factorials, permutation, and/or combinations, these are
the steps to use the functions:
Formulas
Say:
n is the number of items (n items)
c and d are the number of duplicate items
n! is read as n-factorial
The number of permutations of nitems is n!
The number of permutations of duplicate items is $\dfrac{n!}{c! * d!}$
The number of permutations of $n$ total items taking $r$ items at a time is $^nP_r \;\;\;or\;\;\; _nP_r
\;\;\;or\;\;\; P(n, r)$
The number of combinations of $n$ total items taking $r$ items at a time is $^nC_r \;\;\;or\;\;\; _nC_r
\;\;\;or\;\;\; C(n, r) \;\;\;or\;\;\; \displaystyle{\binom{n}{r}}$
$
(1.)\:\: 0! = 1 \\[3ex]
(2.)\:\: n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 1 \\[3ex]
(3.)\;\; n! = n * (n - 1)! \\[3ex]
(4.)\;\; n! = n * (n - 1) * (n - 2)!...among\;\;others \\[3ex]
(5.)\;\; (n - 1)! = (n - 1) * (n - 2)!...among\;\;others \\[3ex]
(6.)\;\; (n - 2)! = (n - 2) * (n - 3) * (n - 4)!...among\;\;others \\[3ex]
(7.)\;\; (n - 3)! = (n - 3) * (n - 4)!...among\;\;others \\[3ex]
(8.)\:\: P(n, r) = \dfrac{n!}{(n - r)!} \\[5ex]
(9.)\:\: C(n, r) = \dfrac{n!}{(n - r)!r!} \\[5ex]
(10.)\;\; P(n, r) = n! * C(n, r) \\[3ex]
(11.)\;\; C(n, r) = C(n, n - r) \\[3ex]
(12.)\;\; (n - r) * P(n, r) = P(n, r + 1) \\[3ex]
(13.)\;\; Number\;\;of\;\;circular\;\;permutations = (n - 1)! \\[3ex]
$
Case 1:
Given: a certain number of digits/letters say p
(14.) The number of unique number of digits/letters say c digits/letters that can be formed if the
digits/letters may be repeated is $p^c$ digits/letters.
(15.) The number of unique number of digits/letters say c digits/letters that can be formed if the
digits/letters may not be repeated is $P(p, c)$ digits/letters.
Case 2:
Given: a certain number of people or items in a linear random order say $n$
(16.) The number of ways in which two people or two items must be close together is $2 * (n - 1) * (n - 2)!$
ways
She will order materials from a website that offers 4 different print designs and 4 different color schemes that can be used with each design.
The top and bottom edges of Josie’s current lampshade are parallel circles with diameters of length 6 inches and 8 inches, as pictured below.
The centers of the 2 circles are directly above and below one another.
In order to design her lampshade, Josie must select 1 print design and 1 color scheme from the website.
She will also choose 1 fabric: burlap, linen, or silk.
How many different possible lampshade designs does she have to choose from?
$ F.\;\; 7 \\[3ex] G.\;\; 11 \\[3ex] H.\;\; 12 \\[3ex] J.\;\; 24 \\[3ex] K.\;\; 48 \\[3ex] $
Given:
4 different print designs
4 different color schemes
1 fabric: burlap, linen, or silk: 3 materials
To Select:
1 print design
1 color scheme
1 fabric
The number of different possible lampshade designs does she have to choose from is:
$ C(4, 1) \cdot C(4, 1) \cdot C(3, 1) \\[3ex] \text{Because } C(n, 1) = n \\[3ex] \implies \\[3ex] = 4 \cdot 4 \cdot 3 \\[3ex] = 48\;designs \\[3ex] $ Student: How do you know that $C(n, 1) = n$?
Is it based on the formula?
Teacher: Yes, it is based on the formula.
Student: Can you show this calculation using the formula?
Teacher: Sure, let's do it.
$ C(n, r) = \dfrac{n!}{(n - r)!r!} \\[5ex] C(4, 1) \cdot C(4, 1) \cdot C(3, 1) \\[3ex] = \dfrac{4!}{(4 - 1)!1!} \cdot \dfrac{4!}{(4 - 1)!1!} \cdot \dfrac{3!}{(3 - 1)!1!} \\[5ex] = \dfrac{4!}{3! \cdot 1!} \cdot \dfrac{4!}{3! \cdot 1!} \cdot \dfrac{3!}{2! \cdot 1!} \\[5ex] = \dfrac{4 \cdot 3!}{3! \cdot 1} \cdot \dfrac{4 \cdot 3!}{3! \cdot 1} \cdot \dfrac{3 \cdot 2!}{2! \cdot 1} \\[5ex] = 4 \cdot 4 \cdot 3 \\[3ex] = 48\;designs $
