mathmath333
  • mathmath333
Counting question
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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mathmath333
  • 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
  • triciaal
|dw:1440601651278:dw|
triciaal
  • triciaal
@Nnesha can you help with this?

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More answers

ganeshie8
  • ganeshie8
you want to solve for \(n\) in : \[2^n-1 = 63\]
mathmath333
  • mathmath333
is the answer 6
ganeshie8
  • ganeshie8
no wait, they made it tricky
ganeshie8
  • ganeshie8
lets do it step by step
mathmath333
  • 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
  • ganeshie8
forget that link, it would be easy if u try it on ur own
ganeshie8
  • ganeshie8
how good are u with pascal triangle ?
mathmath333
  • mathmath333
i just only know how to form a pascal triangle 1 121 1331...
ganeshie8
  • ganeshie8
lets review one important result of pascal triangle quick
ganeshie8
  • 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
  • ganeshie8
see the pattern ?
mathmath333
  • mathmath333
yes
ganeshie8
  • ganeshie8
can you guess the sum of entries in 100000th row ?
mathmath333
  • mathmath333
\(2^{99999}\) ?
ganeshie8
  • 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
  • Astrophysics
.
ganeshie8
  • ganeshie8
|dw:1440602644567:dw|
mathmath333
  • mathmath333
i didnt understand , do u mean i need to start couting from second row
ganeshie8
  • ganeshie8
now we can say that the sum of entries in \(n\)th row is \(2^n\)
mathmath333
  • mathmath333
ok if we count from the second row as row 1
ganeshie8
  • ganeshie8
Exactly, in other words we're starting our count for rows from 0,1,2,...
ganeshie8
  • ganeshie8
instead of 1,2,3,...
mathmath333
  • mathmath333
ok
ganeshie8
  • ganeshie8
so when somebody asks you to look at "3"rd row of pascal triangle, they really mean the 4th row ok
mathmath333
  • mathmath333
yes i get that.
ganeshie8
  • 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
  • mathmath333
sum of 6th row =\(2^{6}\) sum of 7th row =\(2^{7}\)
ganeshie8
  • ganeshie8
good, how is pascal triangle related to combinations/counting ?
mathmath333
  • mathmath333
i don't knw
ganeshie8
  • ganeshie8
suppose you have \(6\) objects how many total ways are there to choose 2 out of them ?
mathmath333
  • mathmath333
6C2
ganeshie8
  • ganeshie8
simplify, what is the number
mathmath333
  • mathmath333
15
ganeshie8
  • ganeshie8
Yes, now look at the 6th row in pascal triangle : |dw:1440603215894:dw|
ganeshie8
  • ganeshie8
suppose you have \(6\) objects how many total ways are there to choose 3 out of them ?
mathmath333
  • mathmath333
20
ganeshie8
  • ganeshie8
you can guess, it is there in pascal triangle : |dw:1440603390646:dw|
mathmath333
  • mathmath333
yes
ganeshie8
  • ganeshie8
\(\large \binom{6}{4}\) is the number locatd at 4th position in 6th row (we count from 0)
ganeshie8
  • ganeshie8
using pascal's triangle, can you tell the value of \(\dbinom{7}{5}\) ?
mathmath333
  • mathmath333
yes it is same as \(\binom{6}{2}=\binom{6}{4}\)
ganeshie8
  • ganeshie8
Yes, it has that symmtry the values in left half are same as the values in right half
mathmath333
  • mathmath333
7th row 5th number or 2nd number
ganeshie8
  • ganeshie8
what is it
ganeshie8
  • ganeshie8
can you highlight the number
mathmath333
  • mathmath333
sry it was sry it was 3ed number 21
ganeshie8
  • ganeshie8
\(\dbinom{7}{5}\) is the 5th element in 7th row : |dw:1440603778591:dw|
ganeshie8
  • ganeshie8
so basically you want to solve : \[\large \dfrac{2^{2n+1}}{2}-1 = 63\]
mathmath333
  • mathmath333
yea u told that

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