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805+ (Hard)|   Combinations|      
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Bunuel
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A is the set of all 4-digit numbers with digits in increasing order an 13 Apr 2022, 07:45
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43% (02:31) correct 57% (02:31) wrong based on 104 sessions

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A is the set of all 4-digit numbers with digits in increasing order and B is the set of all 4-digit numbers with digits in descending order. The difference between the number of elements in sets A and B is

A. 210
B. 126
C. 84
D. 72
E. 26



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C
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chetan2u
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GMAT Focus 1: 735 Q90 V89 DI81
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A is the set of all 4-digit numbers with digits in increasing order an 13 Apr 2022, 23:22
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Bunuel
A is the set of all 4-digit numbers with digits in increasing order and B is the set of all 4-digit numbers with digits in descending order. The difference between the number of elements in sets A and B is

A. 210
B. 126
C. 84
D. 72
E. 26



Are You Up For the Challenge: 700 Level Questions
Show more


Logical conclusion
every decreasing order when reversed will give you an increasing order. ONLY exception is when the decreasing order has 0 in units digit, as reversing will make it a 3-digit number.
Thus number of types 0ABC, where ABC are in increasing order.

Ways ABC can be in increasing order is selecting 3 digits out of remaining 9 digits (1 to 9) = 9C3 = \(\frac{9!}{6!3!}=\frac{9*8*7}{3*2}=84\)


Calculate each way
Ways of arranging digits in decreasing order = choosing 4 digits out of 10 digits. We cannot have 0 in thousands place, and only way 0 can be used in increasing order is left most digit. = 10C4 =\(\frac{10!}{6!4!}=\frac{10*9*8*7}{4*3*2}=210\)
Ways of arranging digits in increasing order = choosing 4 digits out of 9 digits. There is only one way 4 digits can be arranged in increasing/decreasing order. Thus, it becomes selection or combination question = 9C4 =\(\frac{9!}{5!4!}=\frac{9*8*7*6}{4*3*2}=126\)

Difference = 210-126 = 84


C
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ludicrous
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A is the set of all 4-digit numbers with digits in increasing order an 12 Dec 2024, 09:40
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Does increasing or decreasing order mean - STRICTLY increasing or decreasing?
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glagad
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A is the set of all 4-digit numbers with digits in increasing order an 16 Dec 2024, 19:24
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chetan2u - In your logical conclusion part, you have mentioned that ways to choose 3 digits in increasing order is 9C3. I am unable to understand how we reached that conclusion?? When we select three numbers, they need not be in strictly increasing/decreasing order right?

Can you please help understand the solution more?

Thanks
chetan2u
Bunuel
A is the set of all 4-digit numbers with digits in increasing order and B is the set of all 4-digit numbers with digits in descending order. The difference between the number of elements in sets A and B is

A. 210
B. 126
C. 84
D. 72
E. 26



Are You Up For the Challenge: 700 Level Questions


Logical conclusion
every decreasing order when reversed will give you an increasing order. ONLY exception is when the decreasing order has 0 in units digit, as reversing will make it a 3-digit number.
Thus number of types 0ABC, where ABC are in increasing order.

Ways ABC can be in increasing order is selecting 3 digits out of remaining 9 digits (1 to 9) = 9C3 = \(\frac{9!}{6!3!}=\frac{9*8*7}{3*2}=84\)


Calculate each way
Ways of arranging digits in decreasing order = choosing 4 digits out of 10 digits. We cannot have 0 in thousands place, and only way 0 can be used in increasing order is left most digit. = 10C4 =\(\frac{10!}{6!4!}=\frac{10*9*8*7}{4*3*2}=210\)
Ways of arranging digits in increasing order = choosing 4 digits out of 9 digits. There is only one way 4 digits can be arranged in increasing/decreasing order. Thus, it becomes selection or combination question = 9C4 =\(\frac{9!}{5!4!}=\frac{9*8*7*6}{4*3*2}=126\)

Difference = 210-126 = 84


C
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