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# What is the product of all positive odd integers less than 10000?

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Math Expert
Joined: 02 Sep 2009
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What is the product of all positive odd integers less than 10000?  [#permalink]

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16 Dec 2019, 23:43
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Difficulty:

65% (hard)

Question Stats:

61% (02:16) correct 39% (02:04) wrong based on 98 sessions

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What is the product of all positive odd integers less than 10000?

A. $$\frac{10000!}{{(5000!)^2}}$$

B. $$\frac{10000!}{2^{5000}}$$

C. $$\frac{9999!}{2^{5000}}$$

D. $$\frac{10000!}{2^{5000}. 5000!}$$

E. $$\frac{5000!}{2^{5000}}$$

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Re: What is the product of all positive odd integers less than 10000?  [#permalink]

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17 Dec 2019, 00:49
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10000!= 1*2*3*4*.....*9999*10000

10000!= (1*3*5*....*9997*9999) * (2*4*6.......*9998*10000)

10000!= (1*3*5*....*9997*9999) * $$2^{5000}$$(1*2*3.......*4999*5000)

10000!= (1*3*5*....*9997*9999) * $$2^{5000}$$*(5000!)

(1*3*5*....*9997*9999)= $$\frac{10000!}{ 2^{5000}*(5000!)}$$

Bunuel wrote:
What is the product of all positive odd integers less than 10000?

A. $$\frac{10000!}{{(5000!)^2}}$$

B. $$\frac{10000!}{2^{5000}}$$

C. $$\frac{9999!}{2^{5000}}$$

D. $$\frac{10000!}{2^{5000}. 5000!}$$

E. $$\frac{5000!}{2^{5000}}$$

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##### General Discussion
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Joined: 21 Aug 2019
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What is the product of all positive odd integers less than 10000?  [#permalink]

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19 Dec 2019, 06:08
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[quote="Bunuel"]What is the product of all positive odd integers less than 10000?

A. $$\frac{10000!}{{(5000!)^2}}$$

B. $$\frac{10000!}{2^{5000}}$$

C. $$\frac{9999!}{2^{5000}}$$

D. $$\frac{10000!}{2^{5000}. 5000!}$$

E. $$\frac{5000!}{2^{5000}}$$

1. N= 1*3*5*7*9*.......9999
2.M=1*2*3*4*5*6*7*8*9**.....10000
3.A=2*4*6*8*10***10000

So
M/A gives you N
4. M/A= 1*2*3*4*5*6*7*8*9*10..10000/(2*1)(2*2) (2*3) (2*4) (2*5)......(2*5000)
5. 10000!/2^5000(5000!)
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Re: What is the product of all positive odd integers less than 10000?  [#permalink]

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19 Dec 2019, 12:23
[quote="nick1816"]10000!= 1*2*3*4*.....*9999*10000

10000!= (1*3*5*....*9997*9999) * (2*4*6.......*9998*10000)

10000!= (1*3*5*....*9997*9999) * $$2^{5000}$$(1*2*3.......*4999*5000)

10000!= (1*3*5*....*9997*9999) * $$2^{5000}$$*(5000!)

(1*3*5*....*9997*9999)= $$\frac{10000!}{ 2^{5000}*(5000!)}$$

[quote="Bunuel"]What is the product of all positive odd integers less than 10000?

I understand the math you did but how did you realize to do this???
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Re: What is the product of all positive odd integers less than 10000?  [#permalink]

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22 Jan 2020, 01:54
tried with smaller nos
let 10000 ---> 8
then odd nos are 1 3 5 and 7 , which when multiplied give 105.

D results in the same.
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Re: What is the product of all positive odd integers less than 10000?  [#permalink]

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28 Mar 2020, 10:17
Bunuel wrote:
What is the product of all positive odd integers less than 10000?

A. $$\frac{10000!}{{(5000!)^2}}$$

B. $$\frac{10000!}{2^{5000}}$$

C. $$\frac{9999!}{2^{5000}}$$

D. $$\frac{10000!}{2^{5000}. 5000!}$$

E. $$\frac{5000!}{2^{5000}}$$

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Asked: What is the product of all positive odd integers less than 10000?

The product of all positive odd integers less than 10000 $$= 1*3*5*7*....*9999 = \frac{1*2*3*4*.....9999}{2*4*6*...9998} = \frac{9999!}{2^{4999}*4999!} = \frac{10000!}{2^{5000}*5000!} IMO D$$
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What is the product of all positive odd integers less than 10000?  [#permalink]

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29 Mar 2020, 07:29
followed this approach, not sure if it will be applicable to all such ques:

- Take a sample of first 10 integers,

- Product of odd integers from first 10: 9*7*5*3*1 = 63*15

Now, Go through options and make similar scenario:

Option A: $$\frac{10000!}{(5000!)^2}$$ --> Similar to it, $$\frac{10!}{(5!)^2} = 63*4 .... No$$

Same way...Option B...no

Option C. ..no

Option D: $$\frac{10000!}{2^(5000) .5000!}$$ --> Similar to it, $$\frac{10!}{(2)^5.5!} =63*15$$

IMO Option D!
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What is the product of all positive odd integers less than 10000?   [#permalink] 29 Mar 2020, 07:29