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07 Apr 2020, 21:50
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C
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Difficulty:
95%
(hard)
Question Stats:
41% (02:12) correct
59%
(02:50)
wrong
based on 75
sessions
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How many 5 digit numbers can be formed if either
(a) every successive digit exceeds its predecessor, or
(b) every successive digit is smaller than its predecessor.
A. 252
B. 300
C. 378
D. 512
E. 600
(a) every successive digit exceeds its predecessor, or
(b) every successive digit is smaller than its predecessor.
A. 252
B. 300
C. 378
D. 512
E. 600
01 Jan 2024, 01:29
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Khwarizmi
When you select a group of n distinct numbers from k distinct items, they can be arranged in ascending order in only one way, and in descending order also in only one way. For example, consider a group of 5 digits out of 10 to be {8, 3, 1, 5, 6}. This group can be arranged in ascending order in one way: {1, 3, 5, 6, 8} and in descending order also in one way: {8, 6, 5, 3, 1}. Hence, 10C5, giving the number of 5-digit groups possible out of 10, essentially gives the number of 5-digit groups possible out of 10 arranged in ascending. Similarly, to get the number of 5-digit groups possible out of 10 arranged in descending order, we also do 10C5.
There is a caveat though. When considering 5-digit groups in ascending order, we'd also get the groups with 0 in them. Since a 5-digit number cannot begin with 0, we should subtract from 10C5 the number of 5-digit groups that contain 0, which is 9C4 (this will give groups of 0 paired with any other 4 digits from the 9 remaining).
Therefore, the final answer is:
10C5 (the number of 5-digit groups possible out of 10 arranged in descending order) + (10C5 - 9C4) (the number of 5-digit groups possible out of 10 arranged in ascending, which does NOT begin with 0) =
= 252 + (252 - 126) =
= 378.
Answer: C
Below are questions exploiting the same idea:
https://gmatclub.com/forum/if-four-numb ... 95080.html (sub-500 Level PS)
https://gmatclub.com/forum/m27-184482.html (750-800 Level DS)
https://gmatclub.com/forum/if-there-are ... 04654.html (650-700 Level PS)
https://gmatclub.com/forum/a-deck-of-ca ... 21977.html (750-800 Level PS)
Hope this helps.
General Discussion
22 Jun 2025, 05:22
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Bunuel, like the way you subtracted 9c4 from ascending, don't we need to subtract this from descending as well? Or 9c4 was enough for both? I'm confused.
Bunuel
