Practice Questions on Permutation and Combination

The different arrangements of a given number of things by taking some or all at a time, are called permutations.

Examples of Permutation:

All permutations (or arrangements) made with the letters x, y, z by taking two at a time are (xz, xy, yz, yx, zx, zy).

All permutations made with the letters x, y, z taking all at a time are: ( xyz, xzy, yzx, yxz, zxy, zyx)

Number of Permutations:

Number of all permutations of n things, taken r at a time, is given by:
nPr = n(n - 1)(n - 2) ... (n - r + 1) = n!/(n - r)!

Examples:

5P2 = (5 X 4) = 20.
6P3 = (6 x 5 X 4) = 120.

Number of all permutations of n things, taken all at a time = n!.
If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,

such that (p1 + p2 + ... pr) = n.
Then, number of permutations of these n objects is = n!/(p1!).(p2)!.....(pr!)

Combination:

Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.

Examples of Combination:

Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
Note: AB and BA represent the same selection.
All the combinations formed by a, b, c taking ab, bc, ca.
The only combination that can be formed of three letters a, b, c taken all at a time is abc.
Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
Note that ab ba are two different permutations but they represent the same combination.

Number of Combinations:

The number of all combinations of n things, taken r at a time is:
nCr = n!/(r!)(n - r!) = n(n - 1)(n - 2) ... to r factors/r! .
Note:
nCn = 1 and nC0 = 1.
nCr = nC(n - r)

Examples:

I. 10C3 = (10 * 9 * 8)/3 * 2 * 1 = 120
II. 16C14 = 16C(16 - 14) = 16C2 = 16 x 15/2! = 16 x 15 /2 x 1 = 120
Total Number of combination of n things ,r taken at a time where p things will always occur = n-pCr-p.
Total Number of combination of n things ,r taken at a time where p things will never occur = n-pCr.
The number of dividing n distinct things in r different ways is r ⁿ .
Circular permutation of n things = (n-1)!
Total no possible outcomes from a single throw of a perfect die is 6.
When 'n' dice are thrown simultaneously , there will be total of 6 ⁿ outcomes.
Total no possible outcomes from a single toss of a coin is 2.
When 'n' coins are thrown simultaneously , there will be total of 2 ⁿ outcomes.The outcome of each toss in independent of each other.