SOME USEFUL TIPS

(i)     Number of combinations of dissimilar things taken r at a time when p particular things always occur = ^n-pC_r-p.

 Explanation: Here, actually we are making a selection of (r - p) things out of (n - p) things which can be done in ^n-pC_r-p ways.

(ii)    Number of permutations of n dissimilar things taken r at a time when p particular things always occur = ^n-pC_r-pr!

Explanation: Here, number of combinations is the same as above but every combinations of r things can be permutated in ways and that's why total number of permutations = ^n-pC_r-pr!

(iii)    Number of combinations of n dissimilar things taken r at a time when p particular things never occur = ^n-pC_r-p.

Explanation: Here, actual selection of r things is being made out of (n - p) things and that's why total number of selections = ^n-pC_r-p.

(iv)   Number of permutations of n dissimilar things taken r at a time when p particular things never occur = ^n-pC_r-p.

Explanation: Here, number of combinations is the same as above but every selection made of r things can be permutated in ays and therefore then total no. of permutations = ^n-pC_r-p.

(v)    Gap Method: If there are m men and n women (m > n) and they have to sit in a row in such a way that no two women sit together then total no. of such arrangements = ^m+1C_n. m!

Explanation: If we denote men by m and women by w then there are exactly (m + 1) places in which women can be placed such that no two women will be together. This can be done in ^m+1C_n ways. Moreover, m men can be arranged among themselves in m! ways. Therefore, total number of arrangements : ^m+1C_n.m!

w m w m w .................. m w m w

(vi)   String method: Many a times, one may encounter a problem of arranging n number of persons in a row such that m of them is always together. For this we tie all these m persons with a string i.e. we treat them as one, then we have (n - m + 1). Therefore total number of arrangements = (n - m + 1)!.m!.

Never together: If question is to find the number of arrangements such that m out of n are never all together then such total number of arrangements.

Total possible arrangements of a n persons without any restrictions - Total arrangements when m out of them are always together.