I think you're getting a little confused because you're trying so hard to find a common pattern in all the given examples, when sometimes you are better off learning fixed constructions and which form a verb takes. Let's analyse them one by one: 1) 'It's no use (wait) - waiting -Why can't we use "to wait"? In other words it means as I got it right "It's worthless/pointless to wait".' The expression 'it's no use' always takes an -ing form, while 'it's worthless/pointless' can take either, though the infinitive is way more common. Note that there is another similiar expression, 'it's of no use', that requires the infinitive. So, you can say: It's of no use to wait. It's no use waiting. It's pointless/worthless to wait. It's pointless/worthless waiting. http://forum.wordreference.com/threads/its-of-no-use-to-do-or-doing.2409047/ https://www.englishforums.com/English/ItsNoUseTheresNoPointIn/bbrvw/post.htm 2) 'The boys like (play) games, but hate (do) lessons - playing/doing -Why can't we use to play/to do?' We can. 'Like' and 'hate' (as well as 'love' and 'prefer') allow you to use an infinitive or an -ing form. There is little to no distinction between the choices. One is more likely to use the -ing form to talk about something they are pleased to do, while the infinitive gives more the idea of a habit. That's why maybe the -ing form is more appropriate for your exam's question, because it seems to be talking about something the boys are pleased or not with. http://dictionary.cambridge.org/grammar/british-grammar/common-verbs/hate-like-love-and-prefer 3) 'I couldn't help (overhear) what you said - overhearing -The same here: why can't we use infinitive?' 'Can't/couldn't help' is a fixed phrase that always requires the -ing form. There is a similar expression, 'can't/couldn't help but', which, on its turn, requires the verb's base form. So: I couldn't help overhearing what you said. I couldn't help but overhear what you said.

It's no use (wait) - waiting -Why can't we use "to wait"? In other words it means as I got it right "It's worthless/pointless to wait". -> This phrase can use "to wait" or "waiting". However, "waiting" might be more common. The boys like (play) games, but hate (do) lessons - playing/doing -Why can't we use to play/to do? -> This sentence can use the gerund or the infinitive. The boys like to play games, but hate to do lessons. The boys like playing games, but hate doing lessons. Both are correct, however it is common to use the gerund form with verbs that express likes/dislikes. I couldn't help (overhear) what you said - overhearing -The same here: why can't we use infinitive? -> "can't help" is a phrase that requires a gerund. "I couldn't help overhearing what you said." I don't allow (smoke) in my drawing-room - smoking I don't allow my family (smoke) at all - to smoke I used (make) fire by (rub) two sticks together - I read here http://www.really-learn-english.com/used-to-infinitive-gerund.html that we use infinitive when somebody is used to do something but now he/she doesn't do it. And gerund is used when we're accustomed to do something and feel okay about it. In this example it doesn't say that the person doesn't make fire by rubbing two sticks any longer, so why do we use "to" here? -> The use of the gerund you are referring to requires the verb "to be" (for example: "I am used to making fire by rubbing two sticks together" would be with the gerund). "to used to (do something)" is the verbal phrase in the sentence that requires the infinitive form of the verb. With the verb "rub", it is after a preposition, so it needs to be in gerund form. "I used to make fire by rubbing two sticks together."

I used to make fire by rubbing two sticks together. I'm used to making fire by rubbing two sticks together. I'm accustomed to making fire by rubbing two sticks together. I was used to making fire by rubbing two sticks together. I'll become accustomed to making fire by rubbing two sticks together. I didn't use to make fire by rubbing two sticks together. Did you use to make fire by rubbing two sticks together? http://learnenglish.britishcouncil.org/en/quick-grammar/used-infinitive-and-beget-used http://www.macmillandictionary.com/dictionary/british/used-to_1 http://forum.wordreference.com/threads/be-accustomed-to-do-doing.1458699/ http://www.macmillandictionary.com/dictionary/british/be-get-accustomed-to-something Hope it helps!

http://idioms.thefreedictionary.com/can't+help 4) 'I don't allow (smoke) in my drawing-room - smoking I don't allow my family (smoke) at all - to smoke' You either use 'allow + -ing form' without an object, or 'allow someone to (verb)'. So: I don't allow smoking in my drawing room. I don't allow my family to smoke in my drawing room. I don't allow smoking at all. I don't allow my family to smoke at all. http://ell.stackexchange.com/questions/11193/allow-to-infinitive-substantive-verb-ing 5) 'I used (make) fire by (rub) two sticks together - I read here http://www.really-learn-english.com/used-to-infinitive-gerund.html that we use infinitive when somebody is used to do something but now he/she doesn't do it. And gerund is used when we're accustomed to do something and feel okay about it. In this example it doesn't say that the person doesn't make fire by rubbing two sticks any longer, so why do we use "to" here?' You use the infinitive when somebody USED to do something, in the past, not when they ARE USED to doing it in the present. 'Used to (verb)' is made up of the auxiliary verb 'to use' in the past, plus an infinitive verb; therefore, the 'to' that follows 'used' is not part of the phrase, so much so that you have to conjugate 'to use' properly if the sentence requires you to; for example, 'I didn't use to (verb)'. 'Be/become/get used/accustomed to', on the other hand, has 'to' as a fixed part of it, so it requires a noun after it, and for a verb to be a noun, it needs to be in the -ing form. Also, you can conjugate the main verb as well and create sentences in the past or future. And, to wrap it up, 'by' is a preposition, so any verb after it needs to be in the -ing form. So, some combinations for you to grasp them all: