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A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. He can choose the seven questions in 650 ways.
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To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways in which the selection can be made is 5C3 × 20C9.
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There are 3 books on Mathematics, 4 on Physics and 5 on English. How many different collections can be made such that each collection consists of:
| C1 | C2 |
| (a) One book of each subject; | (i) 3968 |
| (b) At least one book of each subject: | (ii) 60 |
| (c) At least one book of English: | (iii) 3255 |
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There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is to be formed. Find:
| C1 | C2 |
| (a) In how many ways committee: can be formed | (i) 10C2 × 19C3 |
| (b) In how many ways a particular: professor is included | (ii) 10C2 × 19C2 |
| (c) In how many ways a particular: lecturer is included | (iii) 9C1 × 20C3 |
| (d) In how many ways a particular: lecturer is excluded | (iv) 10C2 × 20C3 |
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Find the middle term in the expansion of `(2ax - b/x^2)^12`.
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Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
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Find numerically the greatest term in the expansion of (2 + 3x)9, where x = `3/2`.
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The ratio of the coefficient of x15 to the term independent of x in `x^2 + 2^15/x` is ______.
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Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
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If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
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Find the term independent of x in the expansion of `(3x - 2/x^2)^15`
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Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
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Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`
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Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
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Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
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If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
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Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`
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Find n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7th term from the beginning to the 7th term from the end is `1/6`
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If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
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Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
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