Allow Vs Permit Vs Let

Typical mistake: It is not allowed to smoke in the lecture hall. 

1. All and Permit

These words have similar meaning and uses. 'Permit' is a little bit more formal; 'allow' is more common in informal English. Both words can be followed by object + infinitive: 

We do not allow/permit people to soke in the lecture hall. 

When there is no personal object, a gerund (-ing form) is used. 

We do not allow/permit smoking in the lecture hall. 

In passive sentences, personal subjects and gerund subjects are both possible. 

People are not allowed/permitted to smoke in the lecture hall. 

Smoking is not allowed/permitted in the lecture hall. 

The 'impersonal' passive structure cannot be used with 'allow'. 

Allow (but not permit) can be used with adverb particles: 

She wouldn't allow me in. 

Mary isn't allowed out at night. 

2. Let 

'Let' is the least formal of these three words. 'Please allow me to buy you a drink' sounds polite and formal; 'Let me buy you a drink' sounds friendly and informal. 

'Let' is used in a different conversation from 'allow' and 'permit'; it is followed by 'object + infinitive without to. 

Let me help you. 

'Let' is not usually used in the passive; it is replaced by 'allow'. 

I wasn't allowed to pay for the drinks.