Algebra> A student is allowed to select almost n b...

A student is allowed to select almost n books from a collection (2n + 1) books. If the total number of ways in which he can select at least one book is 63, find the value of n.

Amit Saxena , 10 Years ago

Grade upto college level

1 Answers

Navjyot Kalra

Last Activity: 10 Years ago

Hello Student,

Please find the answer to your question

Number of ways in which a student can select at least one and almost n books out of (2n + 1) books is equal to

= ^{2n + 1}C₁ + ^{2n + 1}C₂ + ^{2n + 1}C₃ + . . . . . . . . . . . . . . . . . . . + ^{2n + 1}C_n

= 1/2 [2. ^{2n + 1}C₁+ 2. ^{2n + 1}C₂ + 2. ^{2n + 1}C₃ + . . . . . . . . . . . . . . . . . . . . . . + 2. ^{2n + 1}C_n]

= 1/2 [(^{2n + 1}C₁ + ^{2n + 1}C_2n) + (^{2n + 1}C₂ + ^{2n + 1}C_{2n - 1}) + (^{2n + 1}C₃ + ^{2n + 1}C_{2n – 2}) + . . . . . . . . . . . . + (^{2n + 1}C_n + ^{2n + 1}C_{n + 1})]

[Using ⁿ C_r = ⁿC_{n – r}]

= 1/2 [^{2n + 1}C₁+ ^{2n + 1}C₂ ^{2n + 1}C₃ + . . . . . . . . . . . . . + ^{2n + 1}C_n + ^{2n + 1}C_{n + 1}+ ^{2n + 1}C_{n + 2} + . . . . . . . . . . . . . . + ^{2n + 1}C_2n]

= 1/2 [^{2n + 1}C₀ + ^{2n + 1}C₁ + ^{2n + 1}C₂ + . . . . . . . . . . . . . . . . + ^{2n + 1}C_{2n + 1} – 1 – 1]

= 1/2 [2 ^{2n + 1} – 2] = 2²ⁿ– 1

ATQ, 2²ⁿ – 1 = 63 ⇒ 2²ⁿ = 64 = 2⁶

⇒ 2n = 6 ⇒ n = 3

Thanks
navjot kalra
askIITians Faculty