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 the next time all three bells (P, Q, and R) will ring together, we need to find the least common multiple (LCM) of their 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.

Let us find the LCM of 20, 30, and 50 using prime factorization:
- Prime factorization of 20:

$20=2^2\times 5$

$30=2\times 3\times 5$

$50=2\times 5^2$

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

Calculating the LCM:

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

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

$\text{LCM}=300\text{minutes}$

Thus, the three bells will ring together again after 300 minutes.

Now, let's convert 300 minutes into hours:

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

Since the bells last rang together at 12:00 PM, we add 5 hours to find the next time they will ring together:

$12:00\text{PM}+5\text{hours}=5:00\text{PM}$

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