Question

Hi, I'd want to know the mining and that small difference between let, allow,permit. Thanx a lot

Answer 1

These words have similar meanings. Permit is more formal than allow.

• Smoking is not allowed / permitted.

Both allow and permit can be followed by object + infinitive.

• We do not allow / permit our kids to stay up late.
• We do not allow / permit people to smoke in the house.
• Please allow / permit me to go.

An -ing form is used after allow / permit if there is no personal object.

• We do not allow / permit smoking in the house.

Permit can be used in the passive form with it. Passive forms of allow cannot be used with it.

• It is not permitted to smoke in the house. (NOT It is not allowed to smoke in the house.)

With adverb particles only allow is used.

• She wouldn't allow me in. (NOT She wouldn't permit me in.)

Let is less formal than allow and permit. It is followed by an object + infinitive without to.

• Please allow me to get you a drink. (Quite formal)
• Please let me get you a drink. (Rather informal)

Passive structures with let are not usually possible.

• I wasn't allowed to go. (NOT I wasn't let go.)

Let can be used with adverb particles like in and out.

• She wouldn't let me in.
  
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Answer 2

Fist of all,'Let' is the most comman word inside them.I will give you some examples at the begining.

I wanted to go outside,but my father didnt LET me see my friend

Hotel services didnt ALLOW my cat TO get in. 

Authorities permit  prisioner's family visiting only once a month (more official)