Anurag ,Anil and Anand got a circular chocolate cake from the shop .They cut the cake into 7 unequal pieces . One of them gets only one piece and the other two get equal number of pieces .What is the number of ways in which they could have shared the cake pieces ??

Options

A. 280
B. 360
C. 420
D. 450

Answer

To solve the problem of how Anurag, Anil, and Anand can share the 7 unequal pieces of cake, we can break down the solution into clear steps.

### Step 1: Identify the distribution of pieces
We know that one person will get 1 piece, and the other two will share the remaining 6 pieces equally. Since there are 7 pieces in total, this means that the two remaining people will each get 3 pieces.

### Step 2: Choose who gets the single piece
We have 3 people (Anurag, Anil, and Anand), and we need to choose one of them to receive the single piece. The number of ways to choose 1 person from 3 is given by the combination formula \( C(n, r) \), where \( n \) is the total number of people and \( r \) is the number of people to choose.

\[ \text{Ways to choose 1 person} = C(3, 1) = 3 \]

### Step 3: Distribute the remaining pieces
After one person has taken 1 piece, there are 6 pieces left. The remaining 2 people will each get 3 pieces. Since the pieces are unequal, we need to find the number of ways to distribute these 6 pieces into 2 groups of 3.

The number of ways to choose 3 pieces from 6 is given by:

\[ \text{Ways to choose 3 pieces from 6} = C(6, 3) = 20 \]

### Step 4: Account for the arrangement of pieces
Since the pieces are unequal, the order in which the pieces are assigned matters. For each selection of 3 pieces for one person, the remaining 3 pieces automatically go to the other person. Therefore, we do not need to multiply by any additional arrangements for the second group.

### Step 5: Calculate the total number of ways
Now, we multiply the number of ways to choose who gets the single piece by the number of ways to distribute the remaining pieces:

\[ \text{Total ways} = \text{Ways to choose 1 person} \times \text{Ways to choose 3 pieces from 6} \]
\[ \text{Total ways} = 3 \times 20 = 60 \]

### Final Answer
Thus, the total number of ways in which Anurag, Anil, and Anand could have shared the cake pieces is **60**.

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