The book is giving you some screwy examples. "Let his game be played by him" is correct but weird. You would only say something like that to make very unusual emphasis.
Here are a couple things to know.
How to make the subordinate clause
To make the object of let into the subject of its own clause, you need to put it into the objective case and put the verb into the infinitive. That's why you say:
Let him play his game.
rather than:
Let he plays his game.
Similarly, you would say Let him be helped by us, not Let he is helped by us. However, this sentence is equally as weird as Let his game be played by him.
Two (or three) senses of let
I think what the book is trying to do is teach two different senses of let at the same time that it's teaching you a tricky form of passive voice. Two of the main senses of the word let mean: (1) allow/permit the clause to happen; (2) suggesting or agreeing that "we" do the clause.
A classic example of the allow/permit sense: Let me go! is what a person who is being held against their will says to their captor.
A classic example of suggesting that "we" do something: Let's go! or Let's get started! is what you say when you want to start doing what you and your listener were just talking about doing together. This sense nearly always has us contracted to 's. (It has to be us rather than we because the subject of the clause has to be in the objective case, as above.)
Those are the most common and simplest examples to remember in order to learn the sound of the language. However, those examples can't be converted to a passive form.
Here's a more-realistic example in both active and passive form:
Let Dr. Kildare see the patient.
Let the patient be seen by Dr. Kildare.
Possibly your book has confused the suggesting/agreeing sense with a third sense of let, expressing a wish. A classic example of using let to express a wish is: Let peace prevail on Earth. Here's a realistic version of what I think your book is trying to demonstrate:
Let us beat our swords into plowshares.
Let all our swords be beaten into plowshares.
or, passively again, without using let:
May all swords be beaten into plowshares.
I can see why your book might have confused these senses. They really are a big, muddy mess. They are all variations on the basic sense of allow/permit, stretched to mean different things by repeated usage. The wishing sense can often be understood as the allowing sense and the suggesting sense simultaneously, where the request/suggestion is addressed to a deity, like Oh, God, please let there be peace on Earth.
The book answer is kind of weird, and arguably changes the meaning of the sentence. I will add that I have only a vague idea of what active and passive voice mean, and can’t tell you whether you succeeded in using passive voice. I’m a native speaker, so I’ve never needed to know the names of these things.
In the active voice sentence: The sentence is talking to a group of people, each and every one of whom can buy a ticket at the counter.
In the book answer, the ‘By You’ significantly changes the meaning from the first sentence. Without that, it means that there is a counter for ticket sales. With the ‘By you’, it implies that a single person is able to buy tickets for a group at the counter. While you is used for both singular and plural, the context here implies it is singular. If it was addressed to a group, I'd expect 'Each of you' or 'all of you'.
Your answer fixes the problem with the book answer. Instead of ‘By you’, you used ‘By all of you’, which shows that you are still addressing the group. ‘By each of you’ might be a little clearer, but either is acceptable in conversation. You are also fine with your placement of the phrase. Both ‘at this counter’ and ‘by all of you’ are referring to bought, so they can go in either order.
The problem with both your answer and the book answer is that you pluralize tickets. If a person can buy one or more tickets, this is fine. The original sentence doesn't say whether or not you can buy multiple tickets, though.
A better answer would preserve the singular of ticket from the first sentence. "A ticket can be bought by each of you at this counter." Here, you need to use each, because each person is only buying a single ticket, as in the first sentence. If you used 'all' instead of 'each' in this sentence, it would mean the group as a whole could only buy one ticket.
Best Answer
I think the easiest way to read this sentence is to add one simple verb to act as a copulative:
Note that this sentence (or the original headline!) does not in any way indicate who expected India to win! It could be the press, the public at large, the writer of the article or some sports expert that said this in an interview.
Information is added that was not there in the original headline! The headline does not say that India expected to win, it say that India was expected to win. Nothing indicates who expected it!
That is an interesting expectation. It is equivalent to:
The original headline clearly states that expectations exist that India would win the match, your version states that India expect somebody will win.
Both the book and your version are introducing India as the party that expects something, but the headline does not have that information. In my view, both versions are incorrect.
Sentence 2 is more straightforward and I agree with the book version. The original sentence is also a complete sentence, not a headline.
I just realized that the original sentence does not have to be a (newspaper) headline. The answer above is obviously reading it as if the sentence appeared as a headline, but of course India also simply be the subject in a normal active sentence. In that case, yes, India or the India team are indeed the subject, and they do expect to win the match.
The book answer is completely correct in that case, of course.
Your version still leaves the main verb in the sentence (expects) in the active voice. The main idea of the original sentence is India expects something. You have change something into an (almost) meaningless passive construct and you changed the meaning of the sentence!
The original sentence India expected (to win the match) should be changed into the passive as It was expected by India (that they would win the match).
You version does use a passive construction for (to win the match), but normally in these exercises you are expected (pun intended) to change the voice of the main clause of the sentence.
Apart from the fact that you do not change the main clause of the sentence, you also have removed the important information about who was expected to win the match! In your version, if India's opponent won, that would still be according to the expectation.