There are 7 distinguishable rings. The number of possible five - rings arrangements on the four fingers (except the thumb) of one hand (the order of the rings on each finger is to be counted and it is not required that each finger has a ring is equal to

Options

A. 214110
B. 211410
C. 124110
D. 141120

Answer

To solve the problem of arranging 5 distinguishable rings on 4 fingers (excluding the thumb), we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Select 5 Rings from 7**:
We need to choose 5 rings from a total of 7 distinguishable rings. The number of ways to choose 5 rings from 7 can be calculated using the combination formula:
\[ \text{Number of ways to choose 5 rings} = \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} \]
Simplifying this gives:
\[ \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21 \]

**Hint**: Use the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) to find the number of ways to choose rings.

2. **Arrange the 5 Rings on 4 Fingers**:
Each of the 5 selected rings can be placed on any of the 4 fingers. Since the order of the rings on each finger matters, we calculate the number of arrangements as follows:
\[ \text{Number of arrangements} = 4^5 \]
This is because each ring can independently be placed on any of the 4 fingers.

**Hint**: Remember that for each ring, you have multiple choices (in this case, 4 fingers).

3. **Calculate the Total Arrangements**:
Now, we multiply the number of ways to choose the rings by the number of arrangements:
\[ \text{Total arrangements} = \binom{7}{5} \times 4^5 = 21 \times 4^5 \]
First, calculate \(4^5\):
\[ 4^5 = 1024 \]
Now, multiply:
\[ 21 \times 1024 = 21504 \]

**Hint**: When multiplying large numbers, break them down into smaller parts if necessary to simplify calculations.

### Final Answer:
The total number of possible arrangements of 5 rings on 4 fingers is **21504**.