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555-605 Level|   Algebra|      
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Bunuel
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 15 Jun 2020, 09:53
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

25% (medium)

Question Stats:

73% (01:45) correct 27% (01:39) wrong based on 129 sessions

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What is the value of \(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

A. 103
B. 102
C. 101
D. 100
E. 99


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C
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 15 Jun 2020, 09:53
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Bunuel
What is the value of \(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

A. 103
B. 102
C. 101
D. 100
E. 99


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Similar question: https://gmatclub.com/forum/what-is-the- ... 26744.html
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 15 Jun 2020, 10:14
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Bunuel
What is the value of \(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

A. 103
B. 102
C. 101
D. 100
E. 99


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(100!+99!) = ((100x99!)+99!) = ((99!(100+1)) = 99!x101
(100!-99!) = ((100x99!)-99!) = ((99!(100-1)) = 99!x99

Similarly...
(99!+98!) = ((99x98!)+98!) = ((98!(99+1)) = 98!x100
(99!-98!) = ((99x98!)-98!) = ((98!(99-1)) = 98!x98

99!x101x97!x99x95!x97....
99!x99x97!x97x....

The answer is 101.
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 15 Jun 2020, 13:17
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Numerator is (100!+99!)(98!+97!)(96!+95!)...(4!+3!)(2!+1!)
Which deduces to (100+1)99!(98+1)97!.........(4+1)3!(2+1)1!

Denominator is (100!−99!)(98!−97!)(96!−95!)...(4!−3!)(2!−1!)
Which deduces to (100-1)99!(98-1)97!.......(4-1)3!(2-1)1!

Now,
(100+1)99!(98+1)97!.........(4+1)3!(2+1)1!
---------------------------------------------------
(100-1)99!(98-1)97!...........(4-1)3!(2-1)1!

Ultimately in this type of problems Left most number of numerator and Right most number of denominator survive.

Thus, the fraction turns to 101/1;

IMO ans is C.
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 16 Jun 2020, 06:56
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Num =99!(101)*97!(99).........3!(5)*1!(3)
Den =99!(99)*97!(97)...........3!(3)*1!(1)

101

IMO C
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 18 Jun 2020, 00:25
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Bunuel
What is the value of \(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

A. 103
B. 102
C. 101
D. 100
E. 99


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=> (99!*101*97!*99*95!*97........3!*5*1!*3)/(99!*99*97!*97*95!*95.......3!*3*1)
We can see that only 101 would be left rest all with cross each other out.

Answer C.
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GMAT Focus 1: 735 Q90 V89 DI81
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 18 Jun 2020, 22:32
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Bunuel
What is the value of \(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

A. 103
B. 102
C. 101
D. 100
E. 99


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\(\frac{(100! + 99!)(98! + 97!)(96! + 95!)...(4! + 3!)(2! + 1!)}{(100! - 99!)(98! - 97!)(96! - 95!)...(4! - 3!)(2! - 1!)}\)

\(=\frac{99!(100 + 1)97!(98 + 1)95!(96 + 1)...3!( 4+ 1)1!(2 + 1)}{99!(100 - 1)97!(98 - 1)95!(96 - 1)...3!( 4- 1)1!(2 - 1)}\)

\(=\frac{(100 + 1)(98 + 1)(96 + 1)...( 4+ 1)(2 + 1)}{(100 - 1)(98 - 1)(96 - 1)...( 4- 1)(2 - 1)}=\)

\(=\frac{(101)(99)(97)...(5)(3)}{(99)(97)(95)...(3)(1)}=101\)

C
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 10 Sep 2022, 02:33
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Hello Bunuel,

Could you please solve this problem as it consumes lots of my time but I couldn't solve it out
(100!-99!)^100-((99!-98!)^100))/((98!-97!)^100))
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Bunuel
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What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. 10 Sep 2022, 03:15
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patwwf
Hello Bunuel,

Could you please solve this problem as it consumes lots of my time but I couldn't solve it out
(100!-99!)^100-((99!-98!)^100))/((98!-97!)^100))
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What is the value of \(\frac{(100!-99!)^{100} - (99! - 98!)^{100}}{(98! - 97!)^{100}}\) ?

Factor out \((97!)^{100}\) both from the denominator and the numerator:

    \(=\frac{(97!)^{100}*(98*99*100-98*99)^{100} - (97!)^{100}*(98*99 - 98)^{100}}{(97!)^{100}*(98 - 1)^{100}}=\)

Reduce by \((97!)^{100}\):

    \(=\frac{(98*99*100-98*99)^{100} - (98*99 - 98)^{100}}{97^{100}}=\)

Factor out \(98^{100}\) from the the numerator:

    \(\frac{98^{100}*((99*100-99)^{100} - (99 - 1)^{100})}{97^{100}}=\)

    \(=\frac{98^{100}*(99^{100}*99^{100} - 98^{100})}{97^{100}}=\)

    \(=\frac{98^{100}*(99^{200}- 98^{100})}{97^{100}}=\)

Hope it's clear.

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