Another

Typical mistakes: 

We need an other chair. 

I've got other three days holiday. 

1. 'Another' is written as one word. 

We need another chair. 

2. Normally, 'another' is followed by a singular noun: we can say 'another day', but not 'another days'. However, 'another' can be followed by 'few' or a number with a plural noun. 

I've got another three days holiday. 

There's room for another few people in the back of the bus. 

Note that instead of 'another three days, we could say 'three more days', but not (in this sense)'three other days'. 

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.