## Useful Tips for Algebra

SOME USEFUL TIPS

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

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

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-p}C_{r-p}r!

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

(iv)Number of permutations of n dissimilar things taken r at a time when p particular things never occur =^{n-p}C_{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-p}C_{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+1}C_{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+1}C_{n}ways. Moreover, m men can be arranged among themselves in m! ways. Therefore, total number of arrangements :^{m+1}C_{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.

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