Differentiation...

There are infinite, alike, blue, red, green and yellow balls. Find the number of ways to select $$10$$ balls.

Total number of ways of selecting two numbers from the set $${1,2,3,...90}$$ so that their sum is divisible by $$3$$ is

In a certain examination paper, there are $$n$$ question. For $$j = 1, 2....n$$, there are $$2^{n-j}$$ students who answered $$j$$ or more questions wrongly. If the total number of wrong answers is $$4095$$, then the value of $$n$$ is:

The number of $$n$$ digit numbers which consists of the digits $$1$$ & $$2$$ only if each digits is to be used atleast once, is equal to $$510$$  then $$n$$ is equal to

Consider the following statements:
$$S_1: \lim_\limits{x \to 0} \dfrac{[x]}{x}$$ is an indeterminate form (where [.] denotes greatest integer function).
$$S_2: \lim_\limits{x\to\infty}\dfrac{sin(3^x)}{3^x}=0$$
$$S_3: \lim_\limits{x \to \infty}\sqrt{\dfrac{x- sinx}{x+cos^2x}}$$ does not exist.
$$S_4:  \lim_\limits{n\to \infty}\dfrac{(n+2)!+(n+1)!}{(n+3)! }(n \in N=0$$
State, in order, whether $$S_1, S_2, S_3, S_4$$ are true or false

Find the missing number, if same rule is followed in all the three figures.

Observe the pattern carefully
$$11\times11=121$$
$$111\times111=12321$$
$$1111\times1111=\,?$$

A library has $$'a'$$ copies of one book, $$'b'$$ copies of each of two books, $$'c'$$ copies of each of three books, and single copy each of $$'d'$$. The total number of ways in which these books can be arranged in a row is

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is