A student must choose a program of four courses from a menu of courses

Question

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?

(A) 6
(B) 8
(C) 9
(D) 12
(E) 16

Are You Up For the Challenge: 700 Level Questions

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Official Answer

C

(A) 6
(B) 8
(C) 9
(D) 12
(E) 16

Are You Up For the Challenge: 700 Level Questions

zhanbo · Accepted Answer

The four-course program must contain English.
So the student should choose three from a menu of five courses: Algebra, Geometry, History, Art, and Latin.
C(5,3)=C(5,2)=10
This program must contain at least one mathematics course: either Algebra or Geometry.
We need to subtract ineligible choices from 10 options. 
There is only one ineligible choice: History, Art, and Latin
So the answer is 10-1=9, which is answer choice  (C) .

Nidzo · Answer

As English is compulsory it is not needed for calculating the answer. So instead of having to choose 4 in total, we now have 3 - one of which must be a mathematics course.

There are 2 math choices and 3 'other' courses. The two possible cases here are: 2 math courses are selected and 1 'other' course, or 1 math course and 2 'other' courses.

2 math courses selected:  
 Number of ways of selecting 2 math courses from 2:   1 
 Number of ways of selecting 1 'other' course:  3 
 Total ways of selecting 2 math courses and 1 'other' course:   3*1=3

1 math course is selected: 
 Number of ways of selecting 1 math courses from 2:   2 
 Number of ways of selecting 2 'other' courses:   3 
 Total ways of selecting 1 math courses and 2 'other' course:   3*2=6

Total of ways of choosing 1 math course or 2:  
 3+6=9

Answer C

Sroy1 · Answer

pls someone enlighten me what am I missing

As english is already selected so we are keeping it out of the calculation. 
Now from algebra and geometry at least one must be selected. So available option are  2C1. 
Now there are total 4 subjects left ( as we already selected one math sub earlier).
From those 4 subjects we need to choose 2 subjects. So available options are  4C2 .
Hence total available ways  2c2*4C2=  12

Posted from my mobile device

zhanbo · Answer

There are possibilities of duplicates in your approach:
For example, you may select algebra in your 2C1 step, and geometry in your 4C2 step.
OR you may select geometry in your 2C1 step, and algebra in your 4C2 step.

Almichaelz · Answer

I think the trick is to always account for all possible outcomes. My mistake was same as @sroy1's

ThatDudeKnows · Answer

English is one of the six but must be included. We really have five courses and need to pick three of them. That's  \frac{5!}{3!2!} = \frac{5*4}{2} = 10 . That's all the possibilities, so we need to exclude the disallowed cases.

There's only one way to exclude both math courses.

10-1 = 9

Answer choice C.

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A student must choose a program of four courses from a menu of courses

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A student must choose a program of four courses from a menu of courses

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