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# Solution

Everyone gave a different answer. This is important. Now, there are twelve people and the spouses do not shake hands with each other. This means every person can shake hands with at most ten others. Thus the maximum number of hands that can be shaken is ten and minimum number is zero.

Prakash has asked eleven people (including his wife) as to how many persons they had shaken hands with. He received eleven answers and each answer was different. This means the answers he received ranged from zero to ten. Thus, there was a person who had not shaken hands with anybody, a second who had shaken hands with only one person, a third who had shaken hands with two persons, a fourth who had shaken hands with three persons, a fifth who had shaken hands with four persons, a sixth who had shaken hands with five persons, a seventh who had shaken hands with six persons, an eighth who had shaken hands with seven persons, a ninth who had shaken hands with eight persons, a tenth who had shaken hands with nine persons, and an eleventh person who had shaken hands with ten people. Thus, the eleven answers can be represented as the following set: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

For the ease of identifying the persons, let's name them by the number of persons they have shaken hands with. Thus, **Zero**
has not shaken hands with anybody, **One** has shaken hands with one person, **Two** has shaken hands with two persons, and
so on.

Let's consider the case of **Ten** first. **Ten** has shaken hands with ten people. **Ten** has not shaken hands with
his/her spouse. This means the ten people who have shaken hands with **Ten** have shaken hands with at least one
person (**Ten**). The only person not to have shaken hands with anybody, therefore, has to be **Ten's** spouse (remember, everyone
has given a different answer; so, there has to be a **Zero**). So, we have identified one couple - **Ten and Zero**. Let's
represent this couple as **(Ten, Zero)**.

Let's take the case of **Nine** now. **Nine** has shaken hands with nine people. **Nine** has not shaken hands with his/her
spouse and with **Zero** (**Zero** has not shaken hands with anybody). Thus, these nine people with whom **Nine** has shaken hands
have shaken hands at least twice (they have already shaken hands with **Ten**). Who, then, is the person who has shaken hands
with only one person? That person, obviously, has to be **Nine's** spouse. Thus, we have the second couple **(Nine, One)**.

Let's turn our attention to **Eight**. **Eight** has shaken hands with eight people, excluding his/her spouse, **Zero**
(**Zero** has not shaken hands with anybody), and **One** (**One** has shaken hands with only one person - **Ten**).
Thus, the eight people with whom Eight has shaken hands have shaken hands at least thrice (with **Ten**, **Nine** and **Eight**).
Thus the only person who has shaken hands with only two persons has to be **Eight's** spouse. Thus, we have the third couple, **(Eight, Two)**.

Now, we come to **Seven**. **Seven** has shaken hands with seven people. **Seven** has not shaken hands with his/her spouse,
**Zero**, **One** and **Two**. The seven people with whom **Seven** has shaken hands have, thus, shaken hands
at least four times (with **Ten**, **Nine**, **Eight**, and **Seven**). Thus, the only person who could have
shaken hands with only three persons is **Seven's** spouse. So, here we have the fourth couple **(Seven, Three)**.

Now, we come to **Six**. **Six** has shaken hands with six people. **Six** could not have shaken hands with his/her spouse
(spouses do not shake hands with each other), with **Zero** (**Zero** does not shake hands), with **One** (**One** has
already shaken hands with **Ten** and is not going to shake any more hands), with **Two** (**Two** has shaken hands with
**Ten** and **Nine**, and is not going to shake any more hands), and with **Three** (**Three** has completed the
quota of three handshakes by shaking hands with **Ten**, **Nine** and **Eight**). The remaining six persons with whom **Six**
shakes hands, thus, have shaken hands with at least five persons (with **Ten**, **Nine**, **Eight**, **Seven** and
now with **Six**). So the only person who could have shaken hands with exactly four
persons is Six's spouse. Thus, we have the fifth couple, **(Six, Four)**.

We have, thus, determined five couples: **(Ten, Zero)**, **(Nine, One)**, **(Eight, Two)**, **(Seven, Three)**,
and **(Six, Four)**. We are left with **Five** and Prakash himself. **Five** is, therefore, Prakash's wife. So the sixth couple is
**(Prakash, Five)**. Prakash's wife has, therefore, shaken hands with five persons.

With how many persons has Prakash shaken hands with? Now, the number of handshakes by each couple sums up to
ten; thus we have the couples: (Ten, Zero), (Nine, One), (Eight, Two), (Seven, Three) and (Six, Four). Thus, it would be
logical to conclude that the sixth couple should be **(five, Five)**. So, Prakash has shaken hands with five others. Let's
see if this holds. **Ten**, **Nine**, **Eight**, **Seven**, and **Six** have definitely shaken hands
with Prakash (otherwise, they could not have been able to fulfill their quotas).

**Ten** has shaken hands with ten people excluding his/her spouse. Thus, **Ten** has had to shake hands with Prakash.
Similarly, **Nine** has shaken hands with nine people excluding his/her spouse and **Zero** (**Zero** is content with "namastes"
and is not shaking hands with anybody). Thus, the remaining nine people with whom **Nine** has shaken hands has to include
Prakash. **Eight** could not have shaken hands with **Zero** (**Zero** is not shaking hands), **One** (**One**
has already shaken hands with **Ten**, and is not shaking any more hands) and **Two** (his/her spouse); the remaining eight persons with whom
**Eight** has shaken hands has to include Prakash. **Seven** could not have shaken hands with **Zero**, **One**,
**Two** and **Three** (his/her spouse); the remaining seven people with whom **Seven** has shaken hands has to include Prakash.
**Six** could not have shaken hands with **Zero**, **One**, **Two**, **Three** and
**Four** (his/her spouse); the remaining six people with whom **Six** has shaken hands has to include
Prakash.

Thus, Prakash has shaken hands with at least five persons - **Ten**, **Nine**, **Eight**, **Seven**
and **Six**.

Prakash could not have shaken hands with **Zero** (**Zero** is content with doing a "namaste" and does not like to shake
hands). He could not have shaken hands with **One** (**One** has already shaken hands with **Ten** and is not going to shake any
more hands. He could not have shaken hands with Two (Two has shaken hands with Ten and Nine, and is not going to shake
any more hands). He could not have shaken hands with **Three** (**Three** has shaken hands with **Ten**, **Nine** and
**Eight**, and is not going to shake any more hands). He could not have shaken hands with **Four** (**Four** has shaken hands
with **Ten**, **Nine**, **Eight** and **Seven**). And, of course, Prakash cannot have shaken hands with **Five**,
his wife. So, Prakash, like a good husband, has shaken hands with five persons just like his wife.

Did I get that right? Or, was I wrong as always?

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- Union Public Service Commission - www.upsc.gov.in
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- Indian Statistical Institute - www.isical.ac.in
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- Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
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- Indian Institute of Information Technology, Allahabad (Deemed University) - www.iiita.ac.in
- Central Marine Fisheries Research Institute, Kochi - www.cmfri.com
- Tata Institute of Social Sciences, Mumbai - www.tiss.edu