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The coefficient of x³ y⁴ z⁵ in the expansion of (xy + yz + xz)⁶ is

1) 70 2) 60 3) 50 4) None of these Solution: (2) 60 \(\begin{array}{l}\text \ We \ know \ general... View Article

In the expansion of (1 + 3x + 2x²)⁶ is , the coefficient of x¹¹ is

1) 144 2) 288 3) 216 4) 576 5) 3 . 211 Solution: (4) 576 \(\begin{array}{l}\text \ General \ term... View Article

The number of terms in the expansion of (a + b + c)¹⁰ is

1) 11 2) 21 3) 55 4) 66 Solution: (4) 66 \(\begin{array}{l}\text \ We \ know \ if \ (a+b+c)^{n}... View Article

If a_k is the coefficient of x^k in the expansion of (1 + x + x²) for k = 0, 1, 2, … 2n then the value of ∑_k=1²ⁿ ka_k is

1) – a0 2) 3n 3) n . 3n+1 4) n . 3n Solution: (2) 3n \(\begin{array}{l}\left(1+x+x^{2}\right)^{n}= a_{0} a_{1} x+a_{2}... View Article

If (1 + 2x + 3x²)¹⁰ = a₀ + a₁ x + a₂x² + …+ a₂₀ x²⁰, then a₁ equals

1) 10 2) 20 3) 210 4) None of these Solution: (2) 20 \(\begin{array}{l}\left(1+2 x+3 x^{2}\right)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots a_{20} x^{20}\\... View Article

²⁰C₄ + 2 . ²⁰C₃ + ²⁰C₂ – ²²C₁₈ is equal to

1) 0 2) 1242 3) 7315 4) 6345 Solution: (1) 0 \(\begin{array}{l}20_{c_{4}}+220_{c_{3}}+20_{c_{2}}-22_{c_{18}} \\ =20_{c_{4}}+20_{c_{3}}+20_{c_{3}}+20_{c_{2}}-22_{c_{18}} \\ =21_{c_{4}}+21_{c_{3}}-22_{c_{18}} \\ =22_{c_{4}}-22_{c_{18}} \\ =0\end{array}... View Article

If n is an integer greater than 1, then a – ⁿC₁ (a – 1) + ⁿC₂ (a – 2) + …+ (- 1)ⁿ (a – n) is equal to

1) a 2) 0 3) a2 4) 2n Solution: (2) 0 \(\begin{array}{l}\begin{array}{l} \text { Rewrite the given series as }\\... View Article

The sum of the coefficients in the expansion of (1 + x – 3x²)³¹⁴⁸ is

1) 8 2) 7 3) 1 4) – 1 Solution: (3) 1 Put x = 1 in the given expansion,... View Article

If C₁, C₂, C₃ are the usual binomial coefficients and S = C₁ + 2C₂ + 3C₃ + …+ nC_n, then S equals

1) n 2n 2) 2n –1 3) n 2n –1 4) 2n + 1 Solution: (1) n 2n \(\begin{array}{l}(1+x)^{n}=C_{0}+C_{1} x+C_{2}... View Article

If the coefficient of rth, (r + 1)th and (r + 2)th term in the binomial expansion of (1 + y)^m are in AP, then m and r satisfy the equation

1) m2 – m(4r – 1) + 4r2 + 2 = 0 2) m2 – m(4r–1) + 4r2 – 2... View Article

³⁰C₀ ³⁰C₁₀ – ³⁰C₁ ³⁰C₁₁ + …… ³⁰C₂₀ ³⁰C₃₀ =

1) 30C11 2) 60C10 3) 30C10 4) 65C55 Solution: (3) 30C10 \(\begin{array}{l}(1-x)^{30}=30_{C_{0}} x^{0}-30_{C_{1}} x+30_{C_{2}} x^{2}+\cdots +(-1)^{30} 30_{C_{30}} x^{30} \ldots(\mathrm{i}) \\... View Article

The value of ⁵⁰C₄ + ∑_r=1⁶ ^56-rC₃ is

1) 56C4 2) 56C3 3) 55C3 4) 55C4 Solution: (1) 56C4 \(\begin{array}{l}\begin{array}{l} 50_{c_{4}}+\sum_{r=1}^{6} 56-r_{C_{3}}=50_{c_{4}}+55_{c_{3}}+54_{c_{3}}+53_{c_{3}}+\cdots 50_{c_{3}} \\ \Rightarrow 50_{C_{3}}+50_{c_{4}}+51_{C_{3}}+52_{c_{3}}+\cdots 55_{C_{3}} \\... View Article

The value of 1² . C₁ + 3² . C₃ + 5² . C₅ + … is

1) n(n – 1)n–2 + n . 2n–1 2) n(n – 1)2n-2 3) n(n –1)2n–3 4) None of these Solution:... View Article

If ¹⁸C₁₅ + 2(¹⁸C₁₆) + ¹⁷C₁₆ + 1 = ⁿC₃, then n is equal to

1) 19 2) 20 3) 8 4) 24 5) 21 Solution: (2) 20 18C15 + 2(18C16) + 17C16 + 1... View Article

If ⁿC₁₂ = ⁿC₆, then ⁿC₂ is equal to

1) 172 2) 153 3) 306 4) 2556 Solution: (2) 153 nC12 = nC6 12 + 6 = n n... View Article

For natural numbers and m and n , if (1 – y)^m (1 + y)ⁿ = 1 + a₁y + a₂y² + … and a₁ = a₂ = 10, then (m , n) is

1) (35 , 20) 2) (45 , 35) 3) (35 , 45) 4) (20 , 45) Solution: (3) (35 ,... View Article

If n = 5, then (ⁿC₀)² + (ⁿC₁)² + (ⁿC₂)² + …+ (ⁿC₅)²

1) 250 2) 254 3) 245 4) 252 5) 258 Solution: (4) 252 \(\begin{array}{l}\begin{array}{l} \left(5 c_{0}\right)^{2}+\left(5_{c_{1}}\right)^{2}+\left(5_{c_{2}}\right)^{2}+\cdots+\left(5_{c_{5}}\right)^{2} \\ 1+5^{2}+10^{2}+5^{2}+1^{2}=252 \\ \therefore\left(5_{c_{0}}\right)^{2}+\left(5_{c_{1}}\right)^{2}+\left(5_{c_{2}}\right)^{2}+\cdots+\left(5_{c_{5}}\right)^{2}\\=252... View Article

The sum of the last eight coefficients in the expansion of (1 + x)¹⁵ is

1) 216 2) 215 3) 214 4) None of these Solution: (3) 214 \(\begin{array}{l}15 c_{0}+15 {c_{1}}+15_{c_{2}}+\cdots+15_{c_{15}}=2^{15}\\ 15_{c_{15}}=15_{c_{0}}, 15_{c_{14}}=15_{c_{1}} \ldots 15_{c_{8}}=15_{c_{7}}\\... View Article

The sum of series ²⁰C₀ – ²⁰C₁ x + ²⁰C₂ x² – ²⁰C₃ + …+ ²⁰C₂₀ x²⁰ is

1) – 20C10 2) (1/2) 20C10 3) 0 4) 20C10 Solution: (2) (1/2) 20C10 \(\begin{array}{l}(1+x)^{20}={ }^{20} C_{0}+{ }^{20} C_{1} x+{... View Article

If C₀, C₁, C₂.., C_n denote the binomial coefficients in the expansion of (1 + x)ⁿ, then C₀ + (C₁ / 2) + (C₂ / 3) + ….. C_n / (n + 1) =

1) [2n+1 – 1] / (n + 1) 2) [2n – 1] / (n + 1) 3) [2n-1 – 1]... View Article

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