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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-pCr-p.
Explanation: Here, actually we are making a selection of (r - p) things out of (n - p) things which can be done in n-pCr-p ways.
(ii) Number of permutations of n dissimilar things taken r at a time when p particular things always occur = n-pCr-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-pCr-pr!
(iii) Number of combinations of n dissimilar things taken r at a time when p particular things never occur = n-pCr-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-pCr-p.
(iv) Number of permutations of n dissimilar things taken r at a time when p particular things never occur = n-pCr-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-pCr-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+1Cn. 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+1Cn ways. Moreover, m men can be arranged among themselves in m! ways. Therefore, total number of arrangements : m+1Cn.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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