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The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidates

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The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidates

by BTGmoderatorDC » Sun Nov 01, 2020 4:34 pm

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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


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Re: The host of a television debate show has to select a 4-member discussion panel out of a list of 22 willing candidate

by deloitte247 » Fri Nov 06, 2020 5:40 pm

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To select a 4-member panel
Total candidates = 22
No of politicians = 5
No of Business men = 6
There are at least four professions from total candidates
no 2 member of paid are to be of the same profession

Target question
In how many ways can the panel be constituted?

Statement 1
The last includes 5 journalists and 2 authors
Politicians are 5 Businessmen = 6 Journalist = 5 Author 2
5+6+5 and 22-18=4 and we still have candidates with unknown profession . Target question cannot be evaluated and statement 1 is NOT SUFFICIENT.
'
Statement 2
The list includes only one profession in which there are fewer than 3 candidates.
Politicians and businessmen have more than three candidates less than three and greater than 3 or greater than three and less than three , then exact number is unknown target question cannot be evaluated and statement 2 is INSUFFICIENT.

combining the both statements together
from statement 1, Journalist = 5, Author = 2
from statement 2, There is only one profession with less than three candidates .
So, m totality , there are 5 politician, 6 Businessmen, 5 Journalists, 2 Authors . Some Authors are already less than three and the remaining 4 have to be from the same profession.

Therefore,
No of ways to select 4 different candidates from 5 different profession without without repetition
$$5C_4=\frac{5}{4!\left(5-4\right)!}=\frac{5!}{4!\cdot1}=\frac{\left(5\cdot4\cdot3\cdot2\cdot1\right)}{4\cdot3\cdot2\cdot1}=5\ \left(both\ statement\ combined\ together\ are\ SUFFICIENT\right)$$ $$Answer\ is\ Option\ C$$
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