Each of the different groups or selections which can be made by taking some or all of a number of things as a time (irrespective of the order) is called a combination. By the number of combination of n thing taken r at a time is meant the number of groups of r things which can be formed from the n things. The same is denoted by the symbol nCr
Combination Formula.
Value of nCr: Each combination consists of r different things which can be arranged among them in r! Ways. Therefore, the arrangement for all the nCr combinations is nCr x r!. This equal to the permutations of n different things taken r at a time.
= n! = n(n-1(n-2)…..(n-r+1)
r!(n-r)! r!
Miscellaneous Types of Combinations
Type 1: To find the total number of combinations of n dissimilar things taking any number of them at a time.
1. When all things are different:
Each thing may be disposed of in two ways. It may either be included or rejected.
Therefore the total number of way of disposing of all the things =2x2x2x2...n times = 2n. But this case includes the case in which all the things are rejected.
Hence the total number of ways in which one or more things are taken = 2n-1.
2. When all things are not different
Suppose, out of (p +q +r+.....) things, p are alike of one kind, q alike of second kind, r alike of a third kind, and the rest different.
Out of p things we may take 0, 1, 2, 3.... or p. Hence they may be disposed of in (p+1) ways. Similarly q a like things may be disposed of in (q +1) and r things in (r+1) ways. The‘t’ different things may be disposed in 2t ways. This includes that case in which all are rejected.
Type2: Division into groups:
To find the number of ways in which p +q different things can be divide into two groups containing p and q things respectively.
Type 3: Permutation and combinations occurring simultaneously.
Combination Formula.
Value of nCr: Each combination consists of r different things which can be arranged among them in r! Ways. Therefore, the arrangement for all the nCr combinations is nCr x r!. This equal to the permutations of n different things taken r at a time.
= n! = n(n-1(n-2)…..(n-r+1)
r!(n-r)! r!
Miscellaneous Types of Combinations
Type 1: To find the total number of combinations of n dissimilar things taking any number of them at a time.
1. When all things are different:
Each thing may be disposed of in two ways. It may either be included or rejected.
Therefore the total number of way of disposing of all the things =2x2x2x2...n times = 2n. But this case includes the case in which all the things are rejected.
Hence the total number of ways in which one or more things are taken = 2n-1.
2. When all things are not different
Suppose, out of (p +q +r+.....) things, p are alike of one kind, q alike of second kind, r alike of a third kind, and the rest different.
Out of p things we may take 0, 1, 2, 3.... or p. Hence they may be disposed of in (p+1) ways. Similarly q a like things may be disposed of in (q +1) and r things in (r+1) ways. The‘t’ different things may be disposed in 2t ways. This includes that case in which all are rejected.
Type2: Division into groups:
To find the number of ways in which p +q different things can be divide into two groups containing p and q things respectively.
Type 3: Permutation and combinations occurring simultaneously.
No comments:
Post a Comment