A group of p politicians will stand in a straight line for a photograp

Question

A group of p politicians will stand in a straight line for a photograph before dining at a circular table. Which of the following represents the number of arrangements possible for all p politicians to be arranged for the photograph and then around the dinner table?

Please make one selection in each column.

wadhwakaran · Answer

Arrangement of p persons in a straight line = p!
Arrangement of p persons at a circular table = (p-1)!

1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320

We need two consecutive factorials i.e. p and p-1

From the above possible options it seems p!=40320 and (p-1)!=5040
 Therefore, photograph=40320 and dinner table = 5040

Kushchokhani · Answer

Arrangement in a straight line (Photograph)=p!
Arrangement in a circle (Dinner table)=(p-1)!

We need to look at the choices to answer this ques and choose the ones for which the answer options fits the above requirement.
We know,
7!=5040
8!=40320

Hence, p! (Photograph)=8!=40320
And, (p-1)! (Dinner table)=7!=5040

Arm007 · Answer

Given: There are p politicians and since every politician is different from other, everyone should be taken as a unique object while calculating the number of arrangements.

When standing for the photograph, all the p politician will stand in a straight line. Hence the number of possible arrangements are p!. Let's call this as np

When seated for dinner, all the p politician will be seated in a circular line. Hence the number of possible arrangements are (p-1)!. Let's call this ns

So, We should look for a pair in options which satisfy (ns,np) or ((p-1)!,p!) condition.

now lets decode the numbers in the options:
6=3!
120=5!
5040=7!
20160=(8!)/2
40320=8!
80640=(8!)*2

Now only 7! or 5040 and 8! or 40320 satisfies our condition.

Hence # arrangements at dinner table = 5040

Hence # arrangements at Photography = 40320

UtkarshAnand · Answer

Concept: 
No. of ways of  arrangement of p dissimilar things in a straight line = p! 
No. of ways of  arrangement of p dissimilar things in a circle = (p-1)!

Approach:  
We need to  find factorial values of two consecutive numbers.

Now, checking the options from the given table.
 5040 and 40320 fit our description. They are factorials of consecutive numbers 7 and 8 respectively.

Answer: 
The number of  arrangements  possible for all p politicians to be arranged around the  Dinner Table (circular)  =  5040  = 7!  .
The number of  arrangements  possible for all p politicians to be arranged for the  Photograph (straight line)  =  40320   = 8!  .

shruthi219 · Answer

photograph (straight line) = 8! = 40320
Dinner table (circular) = (8-1)! =7! = 5040

joanagmat · Answer

photo: p!
table: (p-1)!

- couldn´t find the correct number

JDLaw · Answer

When arranging in a circular setting for N people, 
Total ways = (n-1)!
and for ways in a row for N people= n!
therefore it is evident that the change in both numbers is the factor n, i.e.
Total ways for N people inn a row=n*(total ways of arranging N people in circular setting)
(n)!=n*(n-1)!

Lets just note down the factorials for first few numbers
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880

From the table above in the stem, Only 2 of the numbers match which are factorials for consecutive numbers,
7!=5040 & 8!=40320
so, N=8 people (p=8 polititians)
There for selecting the answers,
For circular=5040
For line=40320

Shankey94 · Answer

So basically here we have two set of arrangements:

One is straight line: photograph
Another is circular : dining

For straight line configuration: 
number of possible arrangements is n!
or in simple words the first person will have n options, the next in line will have n-1 and so on
so total arrangements = n*n-1*n-2....3.2.1 = n!

Looking at the options:
6 = 3!
120 = 5!
5040= 7!
20160 = 4*7!
40320 = 8!
80640 = 2*8!
Since the number we are looking at is a perfect factorial. Hence, except D and F all are not possible

For circular configuration:
the total possible arrangements = n-1!
or the first person will have n options to sit, the second will have n-1 options to sit and so on.
Total ways = n*n-1*n-2*.....4.3.2.1 = n!
The only difference here from straight configuration is that we do not have a starting and ending positions. Every place is similar. Hence, every scenario will be repeated n times
so total possible arrangements = n!/n = n-1!

So combining both the solutions we get
n! = 40320 and n-1! = 5040
hence n = 8

Archit3110 · Answer

ways to arrange P people in line
n! and in circle (n-1)!
Here P has to be 8 
  Dinning table 7! :5040
Photograph 8! ; 40320

av1901 · Answer

Group of 'p' politicians:

1) Stand in a straight line (and then)
2) Dine at a circular table

We know that

1) No. of arrangements of 'p' politicians standing in a straight line =  p! 
2) No. of arrangements of 'p' politicians dining at a circular table =  (p-1)!

So, as obvious, the values depend on the value of 'p' but amongst the options, we need to find two values which correspond to factorial values of two consecutive integers. Let us examine the options

6 = 3!
120 = 5!
 5040 = 7! 
20160 = Not the factorial of an integer
 40320 = 8! 
80640 = Not the factorial of an integer

So, as we can see, we have a 7! and a 8! which are the factorials of consecutive integers

Answer:

Photograph = 8! = 40320
Dinner Table = 7! = 5040

pintukr · Answer

For 'p' politicians,

Linear arrangements possible = p!
Circular arrangements possible = (p-1)!

For the given values ,

3! =6
5! =120
6! =720
7! =5040
8! =40320
9! =321880

Hence, p can be 8 politicians; then

number of arrangements possible for all p politicians to be arranged for the circular dinner table = 7! = 5040
number of arrangements possible for all p politicians to be arranged for the photograph = 8! = 40320

Sajjad1994 · Answer

Hello Everyone!

The OA to this question is:

Dinner Table:  5040
 Photograph:  40320

The same is posted in the main question's post. Almost everyone did a good job.

Just for your information:  Explanations posted after 8 am Pacific time are not included for the competition and also those with only the answers and no explanation.

Sajjad1994 · Answer

Official Explanation

Note that the rules for arrangements include:

Ways to arrange n items in a straight line = n! 
Ways to arrange n items in a circle = (n - 1)!

So what this question is really asking for you to do is to find two adjacent factorials, n! and (n - 1)!. The first two factorials in the list you should know: 6 = 3! and 120 = 5!. Since those are not adjacent, that cannot be a pairing. And since you know that 5! is an answer choice, you should then check for 6!, which is 720 and therefore does not appear.

At this point it may be helpful to look at the remaining answer choices and think in terms of number properties and relative value. Since 10! will end in two zeroes (it has among its unique factors 2, 5, and 10) and ought to have more than five digits, you should see that you're looking for factorials that are either 7!, 8!, and 9!. At this point the math is concise enough that you may just want to multiply those factorials out:

7! = the 6! you already have (720) times 7, which is 5040. 8! = the 5040 you already have times 8, which is 40320.

Since those two are answer choices and fit the description in the question, the answers are 5040 and 40320.

Answer:  
  Dinner Table:  5040
 Photograph:  40320

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A group of p politicians will stand in a straight line for a photograp

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