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# P is the product, indicated above, of all the numbers of the form

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P is the product, indicated above, of all the numbers of the form  [#permalink]

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06 May 2016, 08:14
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Question Stats:

76% (01:17) correct 24% (01:55) wrong based on 362 sessions

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$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$

P is the product, indicated above, of all the numbers of the form $$1 - \frac{1}{k}$$ where k is an integer from 2 to 50, inclusive. What is the value of P?

A) $$\frac{1}{100}$$

B) $$\frac{1}{50}$$

C) $$\frac{1}{49}$$

D) $$\frac{49}{50}$$

E) $$\frac{99}{100}$$
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Re: P is the product, indicated above, of all the numbers of the form  [#permalink]

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06 May 2016, 10:21
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[quote="Sallyzodiac"]$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$

P is the product, indicated above, of all the numbers of the form $$1 - \frac{1}{k}$$ where k is an integer from 2 to 50, inclusive. What is the value of P?

A) 1/100

B) 1/50

C) 1/49

D) 49/50

E) 99/100

Ans B.
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##### General Discussion
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P is the product, indicated above, of all the numbers of the form  [#permalink]

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06 May 2016, 10:40
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Dear Sallyzodiac and Himanshu9818

There is a pattern if you all can analyze -

Sallyzodiac wrote:
$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$

$$P = (1-\frac{1}{2})$$ = $$\frac{1}{2}$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})$$ = $$(\frac{1}{3})$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})$$ = $$(\frac{1}{4})$$

Can you notice a pattern ?

I think by now all of you have understood that it depends on the value of the last denominator...

So, $$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$ = $$(\frac{1}{50})$$

Hence answer will be (B) ,$$(\frac{1}{50})$$
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Re: P is the product, indicated above, of all the numbers of the form  [#permalink]

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14 Jun 2017, 15:40
Abhishek009 wrote:
Dear Sallyzodiac and Himanshu9818

There is a pattern if you all can analyze -

Sallyzodiac wrote:
$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$

$$P = (1-\frac{1}{2})$$ = $$\frac{1}{2}$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})$$ = $$(\frac{1}{3})$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})$$ = $$(\frac{1}{4})$$

Can you notice a pattern ?

I think by now all of you have understood that it depends on the value of the last denominator...

So, $$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$ = $$(\frac{1}{50})$$

Hence answer will be (B) ,$$(\frac{1}{50})$$

So does this mean that if k was from 2 to 10 inclusive, the answer would be 1/10? I'm having a hard time following the math on this one.
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Re: P is the product, indicated above, of all the numbers of the form  [#permalink]

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06 Jul 2017, 17:24
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Sallyzodiac wrote:
$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$

P is the product, indicated above, of all the numbers of the form $$1 - \frac{1}{k}$$ where k is an integer from 2 to 50, inclusive. What is the value of P?

A) $$\frac{1}{100}$$

B) $$\frac{1}{50}$$

C) $$\frac{1}{49}$$

D) $$\frac{49}{50}$$

E) $$\frac{99}{100}$$

Let’s calculate some values from our set to find a pattern:

1 - ½ = ½

1 - ⅓ = ⅔

1 - ¼ = ¾

1 - ⅕ = ⅘

Multiplying these values together, we have:

(1/2)(2/3)(3/4)(4/5)

Notice that the fractions simplify and we are left with ⅕.

So, we are left with the numerator from the first fraction and the denominator from the last fraction.

For P, since the last fraction in the set is 1 - 1/50 = 49/50, P would equal 1/50.

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P is the product, indicated above, of all the numbers of the form  [#permalink]

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07 Jul 2017, 11:17
Nathanlambson wrote:
Abhishek009 wrote:
. . .There is a pattern if you all can analyze -

$$P = (1-\frac{1}{2})$$ = $$\frac{1}{2}$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})$$ = $$(\frac{1}{3})$$

$$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})$$ = $$(\frac{1}{4})$$

Can you notice a pattern ?

[The pattern] depends on the value of the last denominator...

So, $$P = (1-\frac{1}{2})(1-\frac{1}{3})(1-\frac{1}{4})...(1-\frac{1}{50})$$ = $$(\frac{1}{50})$$

Hence answer will be (B), $$(\frac{1}{50})$$

So does this mean that if k was from 2 to 10 inclusive, the answer would be 1/10? I'm having a hard time following the math on this one.

Nathanlambson - yes, if k were 2 to 10 inclusive, the answer would be 1/10.

I had a hard time following the math, too. Perhaps how I rewrote the post might help.

1. From original equation, instead of considering all the factors in parentheses, take them two at a time, and do any arithmetic that is inside parentheses.

$$P = (1-\frac{1}{2})(1-\frac{1}{3})$$ =

$$(\frac{1}{2})(\frac{2}{3})$$ = $$\frac{2}{6}$$ = $$\frac{1}{3}$$

2. Repeat, but put the end product from above as the first of your next two factors

$$(\frac{1}{3})(1-\frac{1}{4})$$ = $$(\frac{1}{3})(\frac{3}{4})$$ = $$\frac{1}{4}$$

And again, two at a time, use end product from previous step as first factor:

$$(\frac{1}{4})(1-\frac{1}{5})$$ = $$(\frac{1}{4})(\frac{4}{5})$$ = $$\frac{1}{5}$$

Abhishek009 wrote "[The product of the preceding factors, at any stage, including the last calculation] depends on the value of the last denominator....

In the last step above, denominator is 5. 1 - (1/5) = 4/5

The pattern also demonstrates that after multiplication by previous term, the numerator is 1, and the product's denominator is 5. $$(\frac{1}{4})(\frac{4}{5})$$ = $$\frac{1}{5}$$.

So the pattern is: the end product of all factors at any point including last term = 1 over whatever denominator the nth term has.

You could calculate k = 2 to 10 inclusive, easily, by multiplying only the first two factors, and by inserting the answer to the arithmetic inside all the parentheses. Thus:

$$(1-\frac{1}{2})(1-\frac{1}{3}) = (\frac{1}{2})(\frac{2}{3})$$ = $$\frac{1}{3}$$ ...

$$\frac{1}{3}$$*$$\frac{3}{4}$$*$$\frac{4}{5}$$ *$$\frac{5}{6}$$*$$\frac{6}{7}$$* $$\frac{7}{8}$$ *$$\frac{8}{9}$$*$$\frac{9}{10}$$ =

Start canceling. You get $$\frac{1}{10}$$

Hope it helps.
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Re: P is the product, indicated above, of all the numbers of the form  [#permalink]

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09 Jul 2017, 11:17
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Another way to look at it is to turn P into a factorial.

(1) $$1 - \frac{1}{2} = \frac{1}{2}$$

(2) $$1 - \frac{1}{3} = \frac{2}{3}$$

(3) $$1 - \frac{1}{4} = \frac{3}{4}$$

(4) $$1 - \frac{1}{5} = \frac{4}{5}$$

$$P=\frac{1}{2} * \frac{2}{3} * \frac{3}{4} * \frac{4}{5} * . . . * \frac{49}{50}$$

∴ $$P=\frac{49!}{50!} = \frac{49!}{50*49!} = \frac{1}{50}$$

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Re: P is the product, indicated above, of all the numbers of the form  [#permalink]

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13 Aug 2018, 03:45
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Re: P is the product, indicated above, of all the numbers of the form   [#permalink] 13 Aug 2018, 03:45
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