In how many ways five different rings can be worn in four fingers with at least one ring in each finger?

Options

A. 120
B. 96
C. 20
D. 480

Explanation

To solve the problem of how many ways five different rings can be worn on four fingers with at least one ring on each finger, we can follow these steps:

### Step 1: Choose Rings for Fingers
Since we have four fingers and we need to place at least one ring on each finger, we first need to choose 4 rings out of the 5 available rings.

The number of ways to choose 4 rings from 5 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

In this case:
\[ \binom{5}{4} = \frac{5!}{4! \cdot (5-4)!} = \frac{5!}{4! \cdot 1!} = 5 \]

### Step 2: Arrange the Chosen Rings
Once we have chosen 4 rings, we need to arrange these 4 rings on the 4 fingers. The number of ways to arrange 4 rings is given by the factorial of the number of rings:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 3: Place the Remaining Ring
Now, we have 1 ring left (since we started with 5 rings and used 4). This remaining ring can be placed on any of the 4 fingers. Therefore, there are 4 options for placing this remaining ring.

### Step 4: Calculate the Total Number of Ways
Now, we can calculate the total number of ways to wear the rings by multiplying the number of ways to choose the rings, the number of ways to arrange them, and the number of options for the remaining ring:
\[ \text{Total Ways} = \binom{5}{4} \times 4! \times 4 \]
Substituting the values we calculated:
\[ \text{Total Ways} = 5 \times 24 \times 4 \]
Calculating this gives:
\[ 5 \times 24 = 120 \]
\[ 120 \times 4 = 480 \]

### Final Answer
Thus, the total number of ways to wear the five different rings on four fingers with at least one ring on each finger is **480**.

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