A lady gives a dinner party for five guests. The number of ways in which they may be selected from among nine friends if two of the friends will not attend the party together is

Options

A. 91
B. 112
C. 119
D. none of these

Answer

To solve the problem of selecting 5 guests from 9 friends, where 2 specific friends cannot attend the party together, we can follow these steps:

### Step 1: Calculate the total number of ways to select 5 guests from 9 friends without any restrictions.
We use the combination formula \( nCr = \frac{n!}{r!(n-r)!} \).

Here, \( n = 9 \) (total friends) and \( r = 5 \) (friends to be selected).

\[ 9C5 = \frac{9!}{5!(9-5)!} = \frac{9!}{5! \cdot 4!} \]

### Step 2: Simplify \( 9C5 \).
Calculating \( 9C5 \):

\[ 9C5 = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \]

### Step 3: Calculate the number of ways to select 5 guests where the 2 specific friends (let's call them A and B) are together.
If A and B are together, we can treat them as a single unit. Thus, we now have 8 units to choose from (the combined unit AB and the remaining 7 friends).

We need to select 4 more guests from the remaining 7 friends (since A and B are already counted as one).

\[ 8C4 = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

### Step 4: Simplify \( 8C4 \).
Calculating \( 8C4 \):

\[ 8C4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]

### Step 5: Calculate the number of ways to select 5 guests such that A and B do not attend together.
To find the number of ways in which A and B do not attend together, we subtract the number of ways in which they are together from the total number of ways.

\[ \text{Ways where A and B do not attend together} = 9C5 - 8C4 = 126 - 70 = 56 \]

### Final Answer:
Thus, the number of ways in which the lady can select 5 guests from 9 friends such that A and B do not attend together is **56**.

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