Three bells P, Q and R are rung periodically in a school. P is rung every 20 minutes, Q is rung every 30 minutes and R is rung every 50 minutes. If all the three bells are rung at 12:00 PM, when will the three bells ring together again the next time?

The correct option is 5:00 PM.

To find when the three bells will ring together again, we need to find the least common multiple (LCM) of their individual ringing intervals. The intervals are given as:
- Bell P rings every 20 minutes.
- Bell Q rings every 30 minutes.
- Bell R rings every 50 minutes.

The time when they will all ring together again is the LCM of 20, 30, and 50 minutes. Let us find the prime factorization of each number:

$$20=2^2\times 5$$

$$30=2\times 3\times 5$$

$$50=2\times 5^2$$

To find the LCM, we take the highest power of each prime factor that appears in these factorizations:
- Highest power of 2 is $2^2$.
- Highest power of 3 is $3^1$.
- Highest power of 5 is $5^2$.

Now, calculate the LCM:

$$\text{LCM}(20,30,50)=2^2\times 3\times 5^2$$

$$\text{LCM}=4\times 3\times 25=300\text{minutes}$$

Next, we convert 300 minutes into hours:

$$300\text{minutes}=\frac{300}{60}\text{hours}=5\text{hours}$$

The bells ring together at 12:00 PM. Adding 5 hours to 12:00 PM gives:
12:00 PM + 5 hours = 5:00 PM.

Therefore, the three bells will ring together again at 5:00 PM.