Find quick revision notes of unit Probability and the Binomial Theorem. This includes chapters- Binomial Theorem, Permutation and Combination, and Probability. This quick revision notes is very important for competitive examination as it will help in revising complete unit in very less time. In UPSEE, WBJEE and other engineering entrance examinations some questions are asked directly on formula basis. This revision notes will help you in revising formulae.
Probability and the Binomial Theorem
- Binomial theorem is used for the expansion of any positive integral power of binomial expression.
- In the expansion of (a + b)n, the general term is denoted as Tr + 1
- In the expansion of (a + b)n, the middle terms are:
(a) If n is even, then there is only one middle term which is given by,
- Term independent of x contains no x, hence, find the value of r for which the exponent of x is zero.
Permutation and Combination
- Permutation is the possible arrangements of a collection of things, where the order is important.
- If an event can occur in m different ways, following which another event can occur in n different ways, following which a third event can occur in p different ways, then the total number of occurrence to ‘the events in the given order is m × n × p.
- The number of permutations of n different objects taken r at a time, where
0 < r ≤ n and the objects do not repeat is
- The number of permutations of n different objects taken r at a time, where repetition is allowed, is nr.
- The number of permutations of n objects, where p objects are of the same kind and rest are all different is
- The number of permutations of n objects, where p1 objects are of one kind, p2 are of second kind, ..., pk are of kth kind and the rest, if any, are of different kind is
- The number of circular permutations of n distinct objects is (n − 1)!.
- If clockwise and anti clockwise circular permutations are considered to be same, then it is
- Combination is the possible arrangements of a collection of things, where the order is not important.
- The number of r-combinations of n distinct objects with repetition is
- Number of ways in which at least one object may be selected out of p alike objects of one type q alike objects of second type and r alike object of third type is
(p + 1)(q + 1)(r + 1) − 1
- Number of ways in which at least one object may be selected from n objects where p alike objects of one type q alike objects of second type and r alike object of third type and rest are different is
(p + 1)(q + 1)(r + 1)2n − (p + q + r) − 1
- Number of ways in which n letters can be put in n corresponding envelopes such that no letter goes to correct envelopes is