Counting question

- mathmath333

Counting question

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- mathmath333

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

- triciaal

|dw:1440601651278:dw|

- triciaal

@Nnesha can you help with this?

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## More answers

- ganeshie8

you want to solve for \(n\) in :
\[2^n-1 = 63\]

- mathmath333

is the answer 6

- ganeshie8

no wait, they made it tricky

- ganeshie8

lets do it step by step

- mathmath333

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

- ganeshie8

forget that link, it would be easy if u try it on ur own

- ganeshie8

how good are u with pascal triangle ?

- mathmath333

i just only know how to form a pascal triangle
1
121
1331...

- ganeshie8

lets review one important result of pascal triangle quick

- ganeshie8

```
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

- ganeshie8

see the pattern ?

- mathmath333

yes

- ganeshie8

can you guess the sum of entries in 100000th row ?

- mathmath333

\(2^{99999}\) ?

- ganeshie8

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"

- Astrophysics

.

- ganeshie8

|dw:1440602644567:dw|

- mathmath333

i didnt understand , do u mean i need to start couting from second row

- ganeshie8

now we can say that the sum of entries in \(n\)th row is \(2^n\)

- mathmath333

ok if we count from the second row as row 1

- ganeshie8

Exactly, in other words we're starting our count for rows from 0,1,2,...

- ganeshie8

instead of 1,2,3,...

- mathmath333

ok

- ganeshie8

so when somebody asks you to look at "3"rd row of pascal triangle, they really mean the 4th row ok

- mathmath333

yes i get that.

- ganeshie8

can you tell me the sum of entries in row 6
and also the sum of entries in row 7
|dw:1440602907133:dw|

- mathmath333

sum of 6th row =\(2^{6}\)
sum of 7th row =\(2^{7}\)

- ganeshie8

good, how is pascal triangle related to combinations/counting ?

- mathmath333

i don't knw

- ganeshie8

suppose you have \(6\) objects
how many total ways are there to choose 2 out of them ?

- mathmath333

6C2

- ganeshie8

simplify, what is the number

- mathmath333

15

- ganeshie8

Yes, now look at the 6th row in pascal triangle :
|dw:1440603215894:dw|

- ganeshie8

suppose you have \(6\) objects
how many total ways are there to choose 3 out of them ?

- mathmath333

20

- ganeshie8

you can guess, it is there in pascal triangle :
|dw:1440603390646:dw|

- mathmath333

yes

- ganeshie8

\(\large \binom{6}{4}\) is the number locatd at 4th position in 6th row
(we count from 0)

- ganeshie8

using pascal's triangle, can you tell the value of \(\dbinom{7}{5}\) ?

- mathmath333

yes it is same as \(\binom{6}{2}=\binom{6}{4}\)

- ganeshie8

Yes, it has that symmtry
the values in left half are same as the values in right half

- mathmath333

7th row 5th number or 2nd number

- ganeshie8

what is it

- ganeshie8

can you highlight the number

- mathmath333

sry it was sry it was 3ed number 21

- ganeshie8

\(\dbinom{7}{5}\) is the 5th element in 7th row :
|dw:1440603778591:dw|

- ganeshie8

so basically you want to solve :
\[\large \dfrac{2^{2n+1}}{2}-1 = 63\]

- mathmath333

yea u told that

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