In a party of 30 people eah shakes hands with the others, Number of hand shakes took place in the party is

Options

A. 435
B. 870
C. 900
D. 535

Answer

To find the total number of handshakes that take place in a party of 30 people where each person shakes hands with every other person, we can use the concept of combinations. Specifically, we need to choose 2 people from the 30 to form a handshake.

Here’s the step-by-step solution:

### Step 1: Understand the Problem
We need to calculate the total number of unique handshakes that occur when each of the 30 people shakes hands with every other person.

### Step 2: Use the Combination Formula
The number of ways to choose 2 people from a group of 30 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of people, and \( r \) is the number of people to choose (which is 2 for a handshake).

### Step 3: Substitute the Values
In our case, \( n = 30 \) and \( r = 2 \):
\[ \binom{30}{2} = \frac{30!}{2!(30-2)!} = \frac{30!}{2! \cdot 28!} \]

### Step 4: Simplify the Factorial
We can simplify this expression:
\[ \binom{30}{2} = \frac{30 \times 29}{2 \times 1} = \frac{870}{2} = 435 \]

### Step 5: Conclusion
Thus, the total number of handshakes that took place in the party is **435**.

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