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Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
nCr + 2 · nCr − 1 + nCr − 2 = n + 2Cr.
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There are 10 persons named\[P_1 , P_2 , P_3 , . . . . , P_{10}\]
Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.
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How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if (i) 4 letters are used at a time
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How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used at a time
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How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?
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Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
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How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
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Find the number of permutations of n different things taken r at a time such that two specified things occur together?
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If 35Cn +7 = 35C4n − 2 , then write the values of n.
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Write the number of diagonals of an n-sided polygon.
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Write the expression nCr +1 + nCr − 1 + 2 × nCr in the simplest form.
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Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]
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Write the maximum number of points of intersection of 8 straight lines in a plane.
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Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.
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Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.
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Write the number of ways in which 12 boys may be divided into three groups of 4 boys each.
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Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.
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Find the mean, mode, S.D. and coefficient of skewness for the following data:
| Year render: | 10 | 20 | 30 | 40 | 50 | 60 |
| No. of persons (cumulative): | 15 | 32 | 51 | 78 | 97 | 109 |
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Calculate coefficient of variation from the following data:
| Income (in Rs): | 1000-1700 | 1700-2400 | 2400-3100 | 3100-3800 | 3800-4500 | 4500-5200 |
| No. of families: | 12 | 18 | 20 | 25 | 35 | 10 |
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An analysis of the weekly wages paid to workers in two firms A and B, belonging to the same industry gives the following results:
| Firm A | Firm B | |
| No. of wage earners | 586 | 648 |
| Average weekly wages | Rs 52.5 | Rs. 47.5 |
| Variance of the |
100 |
121 |
| distribution of wages |
(i) Which firm A or B pays out larger amount as weekly wages?
(ii) Which firm A or B has greater variability in individual wages?
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