## mathmath333 one year ago Counting question

1. 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}}

2. triciaal

|dw:1440601651278:dw|

3. triciaal

@Nnesha can you help with this?

4. ganeshie8

you want to solve for $$n$$ in : $2^n-1 = 63$

5. mathmath333

6. ganeshie8

no wait, they made it tricky

7. ganeshie8

lets do it step by step

8. mathmath333

9. ganeshie8

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

10. ganeshie8

how good are u with pascal triangle ?

11. mathmath333

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

12. ganeshie8

lets review one important result of pascal triangle quick

13. 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

14. ganeshie8

see the pattern ?

15. mathmath333

yes

16. ganeshie8

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

17. mathmath333

$$2^{99999}$$ ?

18. 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"

19. Astrophysics

.

20. ganeshie8

|dw:1440602644567:dw|

21. mathmath333

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

22. ganeshie8

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

23. mathmath333

ok if we count from the second row as row 1

24. ganeshie8

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

25. ganeshie8

26. mathmath333

ok

27. ganeshie8

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

28. mathmath333

yes i get that.

29. ganeshie8

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

30. mathmath333

sum of 6th row =$$2^{6}$$ sum of 7th row =$$2^{7}$$

31. ganeshie8

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

32. mathmath333

i don't knw

33. ganeshie8

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

34. mathmath333

6C2

35. ganeshie8

simplify, what is the number

36. mathmath333

15

37. ganeshie8

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

38. ganeshie8

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

39. mathmath333

20

40. ganeshie8

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

41. mathmath333

yes

42. ganeshie8

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

43. ganeshie8

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

44. mathmath333

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

45. ganeshie8

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

46. mathmath333

7th row 5th number or 2nd number

47. ganeshie8

what is it

48. ganeshie8

can you highlight the number

49. mathmath333

sry it was sry it was 3ed number 21

50. ganeshie8

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

51. ganeshie8

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

52. mathmath333

yea u told that