The host of a television debate show has to select a 4-member

Question

The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidates that includes 5 politicians and 6 businessmen. If the list includes candidates from at least 4 professions and no two members of the discussion panel are to be of the same profession, then in how many ways can the panel be constituted?

I. The list includes 5 journalists and 2 authors

II. The list includes only 1 profession from which there are fewer than 3 candidates

I. The list includes 5 journalists and 2 authors

II. The list includes only 1 profession from which there are fewer than 3 candidates

chetan2u · Accepted Answer

The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidates that includes 5 politicians and 6 businessmen. If the list includes candidates from at least 4 professions and no two members of the discussion panel are to be of the same profession, then in how many ways can the panel be constituted?

I. The list includes 5 journalists and 2 authors

II. The list includes only 1 profession from which there are fewer than 3 candidates

We have to find the number of professions and members in them.

I. The list includes 5 journalists and 2 authors
 So 5P, 6B, 5J and 2A, so 18 candidates. But what about remaining 22-18=4 candidates.

II. The list includes only 1 profession from which there are fewer than 3 candidates
5P and 6B .. rest could be 9 and 2 OR could be 4, 5 and 2..
Different answers possible.
Insufficient

Combined...
We know 5P, 6B, 5J and 2A, so 18 candidates. Now we are also told only one profession has fewer than 3 candidates. We know author is 2, so the remaining 4 have to be one profession as 2+2 or 3+1 is not possible..
Sufficient

C

MBAB123 · Answer

chetan2u , what is the best way to solve this question? The way that I came up with is taking 5 different scenarios by excluding each profession once and then adding the possibilities at the end. I understand it's a DA question, but I as I'm a little weak in combinatorics I'm trying to solve this as well.

harshom · Answer

if one were to answer the ways will it be 5C4 * all the different candidates 
ie. 5C4*5*6*5*2*4 ?

S09 · Answer

Why is 3+1 not possible? The question says 'fewer than 3'. So there could be 2 more professions or 1 more profession. Then answer should be E, right?

sanjitscorps18 · Answer

Given information
 
22 candidates = 5 politicians + 6 businessmen + Unknowns

Statement 1: The list includes 5 journalists and 2 authors
 
 22 candidates = 5 politicians + 6 businessmen + 5 journalists + 2 authors + 4 unknowns

No idea about the unknowns.

Not sufficient

Statement 2: The list includes only 1 profession from which there are fewer than 3 candidates 
 
 22 candidates = 5 politicians + 6 businessmen + Unknowns

Possible scenarios

a. 22 candidates = 5 politicians + 6 businessmen + 9 X + 2 Y
b. 22 candidates = 5 politicians + 6 businessmen + 10 X + 1 Y 
c. 22 candidates = 5 politicians + 6 businessmen + 5 X + 5 Y + 5Z
etc...

No idea about the combination.

Combining statement 1 and 2

22 candidates = 5 politicians + 6 businessmen + 5 journalists + 2 authors + 4 unknowns 
and
 The list includes only 1 profession from which there are fewer than 3 candidates

Since we only have 4 unknowns left, we can bucket them as 
i. 2 Profession X and 2 Profession Y
or
ii. 3 Profession X and 1 Profession Y or vice versa
or
iii. 4 Profession X and 0 Profession Y or vice versa

However, since there can be only 1 profession with less than 3 candidates, we only have the option of 4 and 0

Hence, we get
 22 candidates = 5 politicians + 6 businessmen + 5 journalists + 2 authors + 4 Profession X
where X is the last profession.

We can find out the possible combinations since we have fixed professions and their numbers.

IMO Option C

DylanD · Answer

Pre-Work 
Variables: politicians = "P", businessmen = "B", journalists = "J", authors = "A"
Total to choose from = 22 = 5P + 6B + {at least 2 other categories}
Total to select = 4
Condition: Choose at most 1 from each category
Count = (5P)Choose(1P) + (6B)Choose(1B) + (?)Choose(1?) + (?)Choose(1?) + ...

Rephrase the Question : What are the totals of each unknown category?

(1)
i) 5J + 2A = extra 7
ii) Total remaining = 22 - (5P + 6B + 7extra) 
Total of 4 remaining from an unknown number of categories
iii) T = 18 + {4 from one category} OR {3 from 1 category and 1 from another} OR {2 from 1 category and 2 from another category} OR {1 from 4 different categories}
-4 scenarios with different results 
 A,D ]

(2)
i) Be careful not to assume (2) implies (1) and erroneously eliminate A, B, and C
"Fewer than 3" could mean 1 profession has either 1 OR 2 members
-Without (1), (2) results in even more than 4 scenarios with different results
 B ]

(3) = (1) + (2)
i) Applying (2) to (1), only 1 of the 4 scenarios is possible in (1)
This is because the other 3 scenarios involve an additional category containing less than 3 members.
T = 18 + (4 from 1 category)

ANSWER: C

The host of a television debate show has to select a 4-member

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