Counting by cases

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Daniel Mansfield

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Needs to be tested

6 years, 3 months ago

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Daniel Mansfield 6 years, 3 months ago

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Daniel Mansfield 6 years, 3 months ago

Gave some feedback: Needs to be tested

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Daniel Mansfield 6 years, 3 months ago

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Question statement

Suppose you have a hypothetical new alphabet with $\var{c+v}$ letters: $\var{c}$ consonants and $\var{v}$ vowels.

Note that the NUMBAS syntax for $C(n,r)$ is comb(n,r).

Give any introductory information the student needs.

			Name	Type	Generated Value

total	number	759375
c	integer	12
v	integer	3
w	integer	5
answers	list	List of 6 items
ans3	number	510543
A0	number	248832
A1	number	311040
A2	number	155520
A3	number	38880
A4	number	4860
A5	number	243

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Name

Data type

Value

xxxxxxxxxx
 
(c+v)^w

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

759375

← Depends on:

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Advice

(a) The number of words is $(\var{c} + \var{v})^\var{w} = \var{c+v}^\var{w}$ .

(b) The number of $\var{w}$ -letter words with exactly $n$ vowels is

(number of ways of choosing positions of the vowels) $\times$ (number of ways of choosing the vowels) $\times$ (number of ways of choosing the consonants) = $\begin{pmatrix} \var{w} \\ n \end{pmatrix} \times \var{v}^n \times \var{c}^{(\var{w}-n)}$ .

The total is

$\displaystyle \sum\limits_{n=0}^\var{w} \left( \begin{pmatrix} \var{w} \\ n \end{pmatrix} \times \var{v}^n \times \var{c}^{(\var{w}-n)} \right) = (\var{v} + \var{c})^\var{w}$ .

number of words with exactly one vowel + number of words with exactly two vowels + number of words with exactly three vowels + number of words with exactly four vowels + number of words with exactly five vowels = $\var{answers[1]} + \var{answers[2]} +\var{answers[3]} +\var{answers[4]} +\var{answers[5]} = \var{ans3}$ .

Alternatively, it is all the words except those with no vowels

number of words $-$ number of words with exactly zero vowels = $(\var{v} + \var{c})^\var{w} - \var{c}^\var{w} = \var{(v+c)^w - answers[0]}$ .

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This question is used in the following exam:

Week 8: Further counting by Daniel Mansfield in Discrete Mathematics.

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Daniel Mansfield 6 years, 3 months ago

Daniel Mansfield 6 years, 3 months ago

Daniel Mansfield 6 years, 3 months ago

Error in variable testing condition

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Counting by cases

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Daniel Mansfield 6 years, 3 months ago

Daniel Mansfield 6 years, 3 months ago

Daniel Mansfield 6 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `number`