Permutations & Combinations Question - None

Question

If ${}^nC_r = 84$, ${}^nC_{r-1} = 36$, and ${}^nC_{r+1} = 126$, then find the value of n.

Accepted Answer

Step 1: Use the ratio of consecutive binomial coefficients.

From ⁿC_r / ⁿC_(r-1) = (n-r+1)/r:

84/36 = (n-r+1)/r

7/3 = (n-r+1)/r

7r = 3(n-r+1)

7r = 3n - 3r + 3

10r = 3n + 3 ... (equation 1)

Step 2: Use the ratio of the other pair.

From ⁿC_(r+1) / ⁿC_r = (n-r)/(r+1):

126/84 = (n-r)/(r+1)

3/2 = (n-r)/(r+1)

3(r+1) = 2(n-r)

3r + 3 = 2n - 2r

5r + 3 = 2n ... (equation 2)

Step 3: Solve the system of equations.

From equation 1: 10r = 3n + 3 → n = (10r - 3)/3

Substitute into equation 2: 5r + 3 = 2·(10r - 3)/3

15r + 9 = 20r - 6

15 = 5r

r = 3

Step 4: Find n.

From equation 2: 5(3) + 3 = 2n

18 = 2n

n = 9

Verification: ⁹C₃ = 84 ✓, ⁹C₂ = 36 ✓, ⁹C₄ = 126 ✓

∴ Answer: 9

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