a student is allowed to select at most n books from a collection of 2n+1 books . if the total no of ways in which he can select is 63.find n

Here, the student has to select at most n books out of total (2n + 1) books. So, he may choose may 1, 2, 3, or maximum n books.

So, the total number of ways in which at least 1 book is to be select are:

²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ²ⁿ⁺¹C₃ + ..... + ²ⁿ⁺¹C_n = 63 = x(say)  

We know that, ⁿC₀ + ⁿC₁ + ⁿC₂ + .... + ⁿC_n = 2ⁿ 

Therefore, 

²ⁿ⁺¹C₀ + ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + ..... + ²ⁿ⁺¹C_n +  ²ⁿ⁺¹C_2n+1 = 2²ⁿ⁺¹ ..... (1)

Now, we have  ²ⁿ⁺¹C₀ = ²ⁿ⁺¹C_2n+1 = 1 

 and ²ⁿ⁺¹C₁ = ²ⁿ⁺¹C_2n [using ⁿC_r = ⁿC_n-r  ] etc...

On putting the above values in (1), we get

1 + 1 + 2x = 2²ⁿ⁺¹ ..

⇒ 2 + 2(63) = 2²ⁿ⁺¹ .

⇒ 2²ⁿ⁺¹ =  2 + 126 = 128 

⇒ 2²ⁿ⁺¹ =  2⁷ 

⇒ 2n + 1 = 7  [on comparing]

⇒ 2n = 6

⇒ n = 3