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(2) If the coefficients of (2r + 4)th, (...

(2) If the coefficients of (2r + 4)th, (r - 2)th terms in the expansion of `(1 + x)^18` are equal, find r.

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Given expansion is `(1 + x)^(18)`
Now, `(2r + 4)` th term i.e., `T_(2r + 3 + 1)`
`:. T_(2r+ 3 + 1) = .^(18)C_(2r + 3) (1)^(18 - 2r - 3) (x)^(2r + 3)`
`= .^(18)C_(2r + 3) x^(2r + 3)`
Now, `(r - 2)` th term i.e., `T_(r - 3+ 1)`
`:. T_(r - 3 + 1) = .^(18)C_(r - 3) x^(2r + 3)`
Now, `(r - 2)th` term i.e., `T_(r - 3 + 1)`,
`:. T_(r - 3 + 1) = .^(18)C_(r - 3) x^(r - 3)`
As, `.^(18)C_(2r + 3) = .^(18)C_(r - 3) [:. .^(n)C_(x) = .^(n)C_(y) rArr x + y = n]`
`rArr 2r + 3 + r - 3 = 18`
`rarr 3r = 18`
`:. r = 6`
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