Last visit was: 15 Jul 2024, 01:54 It is currently 15 Jul 2024, 01:54
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

What is the value of (100! + 99!)(98! + 97!)(96! + 95!)..

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94343
Own Kudos [?]: 640910 [5]
Given Kudos: 85011
Math Expert
Joined: 02 Sep 2009
Posts: 94343
Own Kudos [?]: 640910 [0]
Given Kudos: 85011
Current Student
Joined: 05 Oct 2019
Posts: 100
Own Kudos [?]: 28 [0]
Given Kudos: 9
Location: India
Concentration: Finance, Accounting
GMAT 1: 680 Q46 V38
GMAT 2: 770 Q50 V44 (Online)
Manager
Joined: 24 May 2018
Posts: 96
Own Kudos [?]: 100 [0]
Given Kudos: 35
Location: India
Concentration: General Management, Human Resources
WE:Engineering (Energy and Utilities)
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
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.
Senior Manager
Joined: 05 Aug 2019
Posts: 317
Own Kudos [?]: 286 [0]
Given Kudos: 130
Location: India
GMAT 1: 600 Q50 V22
GMAT 2: 670 Q50 V28 (Online)
GPA: 4
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
Num =99!(101)*97!(99).........3!(5)*1!(3)
Den =99!(99)*97!(97)...........3!(3)*1!(1)

101

IMO C
Retired Moderator
Joined: 05 May 2016
Posts: 770
Own Kudos [?]: 695 [0]
Given Kudos: 1316
Location: India
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
Bunuel wrote:
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

Are You Up For the Challenge: 700 Level Questions

=> (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.

RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11471
Own Kudos [?]: 34326 [2]
Given Kudos: 322
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
1
Kudos
1
Bookmarks
Bunuel wrote:
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

Are You Up For the Challenge: 700 Level Questions

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

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

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

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

C
Intern
Joined: 08 Sep 2022
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 1
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
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))
Math Expert
Joined: 02 Sep 2009
Posts: 94343
Own Kudos [?]: 640910 [1]
Given Kudos: 85011
Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
1
Kudos
patwwf wrote:
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))

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.

Re: What is the value of (100! + 99!)(98! + 97!)(96! + 95!).. [#permalink]
Moderator:
Math Expert
94343 posts