Q. By the expression \(\left(\frac{x}{3} + \frac{1}{x}\right)^5\), evaluate the middle term in the expression.

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Q. Evaluate the expression (y+1)⁴ – (y-1)⁴.

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Q. Find the coefficient of x⁷ in (x+4)⁹.

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Q. Determine the 7th term in the expansion of (x-2y)¹².

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Q. What is the middle term in the expansion of (x/2 + 6y)⁸?

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Q. Calculate the value of ⁸C₅.

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Q. In how many ways can you select 9 cupcakes from a box containing 17 cupcakes?

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Q. How many 4-digit numbers can be formed by using 2, 4, 6, 8, 10, 12 without repetition of digits?

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Q. What is the coefficient of x⁹ in the expansion of (x+5)¹⁴?

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Q. Determine the independent term of x⁷ in the expansion of (3x² + 4)¹².

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Q. In a game, a fair coin is tossed 6 times. Each time the coin comes up tails, A will pay Rs. 15 but if each time heads come up, A will pay nothing. Determine the probability that A will win Rs. 45 by playing the game?

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Q. Find the coefficient of x⁸ in the expansion of (x+2)¹¹.

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Q. Determine the coefficient of the x⁵y⁷ term in the polynomial expansion of (m+n)¹².

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Q. The independent term of x is 80000 in the expansion of (3x+b/x)⁶, where b is a positive constant. What the value of b?

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Q. Consider the recurrence relation a₁=4, aₙ=5n+aₙ₋₁. The value of a₆₄ is _________

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Q. Determine the solution of the recurrence relation Fₙ=20Fₙ₋₁ − 25Fₙ₋₂ where F₀=4 and F₁=14.

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Q. What is the recurrence relation for 1, 7, 31, 127, 499?

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Q. If Sₙ=4Sₙ₋₁+12n, where S₀=6 and S₁=7, find the solution for the recurrence relation.

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Q. Find the value of a₄ for the recurrence relation aₙ=2aₙ₋₁+3, with a₀=6.

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Q. The solution to the recurrence relation aₙ=aₙ₋₁+2n, with initial term a₀=2 are _________

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