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Find the unknown value:

(1) nC2 = 28

(2) 27Cr+4 = 27C2r-1

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(1) nC2 = 28

nC2 = 28

∴ \(\frac{n!}{2!(n−2)!}\) = 28

∴ \(\frac{n(n−1)(n−2)!}{2×(n−2)!}\) = 28

∴ n(n – 1) = 28 × 2

∴ n(n – 1) = 56

∴ n(n – 1) = 8 × 7

∴ n(n – 1) = 8(8 – 1)

∴ n = 8

(2) 27Cr+4 = 27C2r-1

27Cr+4 = 27C2r-1

nCx = nCy

Or

n = x + y

27Cr+4 = 27C2r-1

∴ r + 4 = 2r – 1

∴ 4 + 1 = 2r – r

∴ r = 5

Or

r + 4 + 2r – 1 = 27

∴ 3r + 3 = 27

∴ 3r = 27 – 3

∴ r = 24 / 3 = 8

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