If (^n C2)/(^n P3) = 1/10, find the value of n.

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Here are the solutions for questions 2, 3, and 4. Question 2: a) If (^n C₂)/(^n P₃) = 1/10, find the value of n. Step 1: Write out the formulas for combinations and permutations. ^n C_r = (n!)/(r!(n−r)!); ^n P_r = (n!)/((n−r)!) Step 2: Apply the formulas to ^n C₂ and ^n P₃. ^n C₂ = (n!)/(2!(n−2)!) = (n(n−1)(n−2)!)/(2 · 1 · (n−2)!) = (n(n−1))/(2); ^n P₃ = (n!)/((n−3)!) = (n(n−1)(n−2)(n−3)!)/((n−3)!) = n(n−1)(n−2) Step 3: Substitute these expressions into the given equation. ((n(n−1))/(2))/(n(n−1)(n−2)) = 1/10 Step 4: Simplify the equation. Assuming n ≠ 0 and n ≠ 1 (which must be true for the terms to be defined, as n ≥ 3 for ^n P₃), we can cancel n(n−1) from the numerator and denominator. (1)/(2(n−2)) = 1/10 Step 5: Solve for n. 2(n−2) = 10; n−2 = 10/2; n−2 = 5; n = 5+2; n = 7 The value of n is: 7 b) Evaluate log_p 2 − log_p 9 + log_p 4√(3). Step 1: Use the logarithm properties log a − log b = log a/b and log a + log b = log (ab). log_p 2 − log_p 9 + log_p 4√(3) = log_p (2/9) + log_p 4√(3) Step 2: Combine the terms. log_p (2/9 × 4√(3)) log_p ((8√(3))/(9)) The evaluated expression is: log_p ((8√(3))/(9)) Question 3: The product of the 3rd and 7th terms of an arithmetic progression is 105. The 18th term divided by the 5th term is 5. The first term is positive. Find: a) second term b) Sum of the first 20 terms. Step 1: Define the terms of an arithmetic progression. Let the first term be a and the common difference be d. The n-th term is T_n = a + (n−1)d. T₃ = a+2d; T₇ = a+6d; T₅ = a+4d; T₁₈ = a+17d Step 2: Set up equations based on the given information. 1) Product of 3rd and 7th terms is 105: (a+2d)(a+6d) = 105 2) 18th term divided by 5th term is 5: (a+17d)/(a+4d) = 5 Step 3: Solve equation (2) for a in terms of d or d in terms of a. a+17d = 5(a+4d); a+17d = 5a+20d; 17d − 20d = 5a − a; −3d = 4a; d = −4/3a Step 4: Substitute the expression for d into equation (1). (a+2(−4/3a))(a+6(−4/3a)) = 105; (a−8/3a)(a−24/3a) = 105; ((3a−8a)/(3))((3a−24a)/(3)) = 105; (−5a/3)(−7a) = 105; (35a²)/(3) = 105 Step 5: Solve for a. 35a² = 105 × 3; 35a² = 315; a² = 315/35; a² = 9; a = ± 3; Since the first term is positive, a=3. Step

If (^n C2)/(^n P3) = (1)/(10), find the value of n.

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Step-by-step solution

Part 2

Part a· 5 steps7

Write out the formulas for combinations and permutations.

Apply the formulas to $^n C_2$ and $^n P_3$ .

Substitute these expressions into the given equation.

Simplify the equation.

Solve for $n$ .

Part b· 2 steps
$\log_p \left(\frac{8\sqrt{3}}{9}\right)$

Use the logarithm properties $\log a - \log b = \log \frac{a}{b}$ and $\log a + \log b = \log (ab)$ .

Combine the terms.

Part 3· 5 steps

Define the terms of an arithmetic progression.

Set up equations based on the given information.

Solve equation (2) for $a$ in terms of $d$ or $d$ in terms of $a$ .

Substitute the expression for $d$ into equation (1).

Solve for $a$ .

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