(Q). A student is allowed to select at most n books from a collection of (2n+1) books. If the total number of ways in which he can select at least one book is 63, then what is the value of n?
Solution:-
total number of ways to select at least 1 & at most n books out of (2n+1) books is
= ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ²ⁿ⁺¹C₃ + ²ⁿ⁺¹C₄ + ……………… + ²ⁿ⁺¹Cₙ ……………….. equation (i)
= 63 (Given)
now ²ⁿ⁺¹C₀ + ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ²ⁿ⁺¹C₃ + ²ⁿ⁺¹C₄ +………….+ ²ⁿ⁺¹Cₙ₋₁ + ²ⁿ⁺¹Cₙ + ²ⁿ⁺¹Cₙ₊₁ + ²ⁿ⁺¹Cₙ₊₂ + ²ⁿ⁺¹Cₙ₊₃ + ………………. + ²ⁿ⁺¹C₂ₙ + ²ⁿ⁺¹C₂ₙ₊₁
= 2²ⁿ⁺¹ …………………… equation(ii)
equation (ii) can again be written as
²ⁿ⁺¹C₀ + ²ⁿ⁺¹C₂ₙ₊₁ + (²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ²ⁿ⁺¹C₃ + ²ⁿ⁺¹C₄ +………….+ ²ⁿ⁺¹Cₙ₋₁ + ²ⁿ⁺¹Cₙ) + (²ⁿ⁺¹Cₙ + ²ⁿ⁺¹Cₙ₋₁ + ²ⁿ⁺¹Cₙ₋₂ + ……………….. + ²ⁿ⁺¹C₁)
= 2²ⁿ⁺¹
⟹ 1 + 1 +2(²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ²ⁿ⁺¹C₃ +………….+ ²ⁿ⁺¹Cₙ) = 2²ⁿ⁺¹
⟹ 2 + 2(63) = 2²ⁿ⁺¹
⟹ 1+ 63 = 2²ⁿ
⟹ 2⁶ = 2²ⁿ
∴ n = 3 Answer
