List of top Mathematics Questions on permutations and combinations asked in AIEEE

Let $a_{n}$ denote the number of all $n$ -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are $0 .$ Let $b_{n}=$ the number of such $n$ -digit integers ending with digit 1 and $c_{n}=$ the number of such $n$ -digit integers ending with digit $0 .$ Which of the following is correct?

If $n= \,^mC_2$, then the value of $^nC_2$ is given by

There are 10 points in a plane, out of these 6 are collinear. If N is the number of triangles formed by joining these points. then :

Let $S_{1} = \sum\limits^{10}_{j = 1}j\left(j-1\right) ^{10}C_{j} , S_{2} = \sum\limits^{10}_{j = 1}j ^{10}C_{j}$ and $S_{3} =\sum\limits^{10}_{j = 1}j^{2} \,^{10}C_{j}.$ $S_3 = 55 \times $2^9$$ $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$.

From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is

The set $S=\left\{1, 2, 3, \dots, 12\right\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus, $A\cup B\cup C=S, A\cap B = B\cap C = A \cap C=\phi.$ The number of ways to partition $S$ is

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number:

The range of the function $f\left(x\right) = ^{7-x}P_{x-3}$ is