5!

It's 6!

The answer is 16!
2people=1handshake
3 =3;
4 =6;
5 =10;
6 =15;
7 =21;
n =nCn-2;
=n(n-1)/2;
now you solve the following equation
n(n-2)/2=120 and you will get 16.
this is from a solid math background, thanks!

Iamdagreatguy:
Weekend brain drill. The first to get the answer wins a prize:

At a birthday party, everyone shook hands with everybody else. There were 120 handshakes. How many people were at the party?

16 people , did it in 16 secs

16

16

61

Craigston:

You're good. I got to this point but the combination gave me a quadratic equation with no real roots:
n ( n - 1 ) / 2 = 120;
n² - n = 240;
n² - n - 240 = 0;

It should be n(n+1)-240=0
n(2+(n-1))=240
n(2+n-1)=240
n(1+n)=20

n+16=0 or n-15=0

But I did (n-1)!= 120 and arrived at n= 5 and the sum the digits in 5! And add one to it.
1+1+2+3+4+5=16

!!!

Non of the above....
wrong question!
Buhari is the answer.
confusion.
No answer.

this doesn't call for any mathematical hypothesis. its simple. 120 persons will shake hands 120 times one shaking hand with every other person. for examine, if a shakes b and c, it's left for b and c to shake since b and c had also shook a. hence we have 3 handshakes from 3 people.

Craigston:

You're good. I got to this point but the combination gave me a quadratic equation with no real roots:
n ( n - 1 ) / 2 = 120;
n² - n = 240;
n² - n - 240 = 0;

dear friend, i think you got the same quadratic equation as mine. but if you get "no real root" then maybe you didn't solve it well! thanks!

Applying permutation.. When there is a circular occurrence the formula is (N - 1)! which is 6 - 1 = 5! Dat is 5x4x3x2x1 = 120...

bollify:
this doesn't call for any mathematical hypothesis. its simple. 120 persons will shake hands 120 times one shaking hand with every other person. for examine, if a shakes b and c, it's left for b and c to shake since b and c had also shook a. hence we have 3 handshakes from 3 people.

what you are saying will only work for 3 people, if you think it will work generally, then you will get it wrong. think about 5 people and you 'll get 10 hand shakes. 6 people will give you 15 hand shakes. so you see?

.

That should help.

The answer is 7

The answer is 5;

5! factorial = 120

Fredoh:
what you are saying will only work for 3 people, if you think it will work generally, then you will get it wrong. think about 5 people and you 'll get 10 hand shakes. 6 people will give you 15 hand shakes. so you see?

Sorry my bad. Thanks for the correction, I now realised it's a series, it follows that if n= the number of people, xn= number of handshakes. Then xn= 0+1+2+......+(n-1) once again, thanks for calling my attention to it.


Modified

Let \(n\) represent the number of people at the party. Each person shakes hands with every other person, and the number of handshakes can be calculated using the formula \( \frac{n \cdot (n-1)}{2} \).

Given that there were 120 handshakes:

\[ \frac{n \cdot (n-1)}{2} = 120 \]

Solving for \(n\):

\[ n \cdot (n-1) = 240 \]

\[ n^2 - n - 240 = 0 \]

Now, factorizing:

\[ (n - 16)(n + 15) = 0 \]

This gives two potential solutions: \(n = 16\) or \(n = -15\). Since the number of people cannot be negative, the only valid solution is \(n = 16\).

Therefore, there were 16 people at the birthday party.


Or


This problem can be solved using the formula for the number of handshakes in a complete graph:

**n * (n - 1) / 2**

where n is the number of people.

We know the number of handshakes (120) and need to solve for n.

Here's how we can find the answer:

1. Set up the equation:

n * (n - 1) / 2 = 120

2. Multiply both sides by 2:

n * (n - 1) = 240

3. Expand the left side:

n^2 - n = 240

4. Factor the quadratic equation:

(n - 16)(n + 15) = 0

5. Solve for n:

n = 16 or n = -15

Since the number of people cannot be negative, the answer is:

**n = 16**

Therefore, there were **16 people** at the birthday party.