Counting question

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\(\large \color{black}{\begin{align} & \normalsize \text{A student is allowed to select at most n books } \hspace{.33em}\\~\\ & \normalsize \text{from a collection of (2n+1) books. if the total number } \hspace{.33em}\\~\\ & \normalsize \text{of ways in which he can select at least one book is 63.find n} \hspace{.33em}\\~\\ & a.)\ 4 \hspace{.33em}\\~\\ & b.)\ 3 \hspace{.33em}\\~\\ & c.)\ 5 \hspace{.33em}\\~\\ & d.)\ 6 \hspace{.33em}\\~\\ \end{align}}\)
|dw:1440601651278:dw|
@Nnesha can you help with this?

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Other answers:

you want to solve for \(n\) in : \[2^n-1 = 63\]
is the answer 6
no wait, they made it tricky
lets do it step by step
here is one of the link i didnt understand the answer given http://www.meritnation.com/ask-answer/question/a-student-is-allowed-to-select-at-most-n-books-from-a-colle/sets/2913008
forget that link, it would be easy if u try it on ur own
how good are u with pascal triangle ?
i just only know how to form a pascal triangle 1 121 1331...
lets review one important result of pascal triangle quick
``` 1 11 121 1331... ``` add the entries in first row, you get : 1 = 1 = 2^0 add the entries in second row, you get : 1+1 = 2 = 2^1 add the entries in third row, you get : 1+2+1 = 4 = 2^2
see the pattern ?
yes
can you guess the sum of entries in 100000th row ?
\(2^{99999}\) ?
Yes, but that looks ugly to have different numbers for index and exponent we can fix it easily, if we say that we start counting the rows from "0"
.
|dw:1440602644567:dw|
i didnt understand , do u mean i need to start couting from second row
now we can say that the sum of entries in \(n\)th row is \(2^n\)
ok if we count from the second row as row 1
Exactly, in other words we're starting our count for rows from 0,1,2,...
instead of 1,2,3,...
ok
so when somebody asks you to look at "3"rd row of pascal triangle, they really mean the 4th row ok
yes i get that.
can you tell me the sum of entries in row 6 and also the sum of entries in row 7 |dw:1440602907133:dw|
sum of 6th row =\(2^{6}\) sum of 7th row =\(2^{7}\)
good, how is pascal triangle related to combinations/counting ?
i don't knw
suppose you have \(6\) objects how many total ways are there to choose 2 out of them ?
6C2
simplify, what is the number
15
Yes, now look at the 6th row in pascal triangle : |dw:1440603215894:dw|
suppose you have \(6\) objects how many total ways are there to choose 3 out of them ?
20
you can guess, it is there in pascal triangle : |dw:1440603390646:dw|
yes
\(\large \binom{6}{4}\) is the number locatd at 4th position in 6th row (we count from 0)
using pascal's triangle, can you tell the value of \(\dbinom{7}{5}\) ?
yes it is same as \(\binom{6}{2}=\binom{6}{4}\)
Yes, it has that symmtry the values in left half are same as the values in right half
7th row 5th number or 2nd number
what is it
can you highlight the number
sry it was sry it was 3ed number 21
\(\dbinom{7}{5}\) is the 5th element in 7th row : |dw:1440603778591:dw|
so basically you want to solve : \[\large \dfrac{2^{2n+1}}{2}-1 = 63\]
yea u told that

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