An Agency forms a code of two distinct digits selected from 0, 1, 2, …

Question

GMATBusters’ Quant Quiz Question -5

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An Agency forms a code of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion, when read upside down-for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?
(a) 69
(b) 71
(c) 78
(d) 80
(e) 89

GMATBusters · Answer

Official Solution

The available digits are 0,1,2, …9.
The first digit can be chosen in 9 ways (0 not acceptable), the second digit can be accepted in 9 ways (digits repetition not allowed).
Thus, the code can be made in 9 × 9 = 81 ways.
Now there are only 4 digits 1, 6, 8, 9 which can create confusion.
Hence, the total number of codes which create confusion are = 4 × 3 = 12.
Out of these 12 codes 69 and 96 will not create confusion.
Hence, in total 12 – 2 = 10 codes will create confusion.
Hence, the total codes without confusion are 81 – 10 = 71.

Answer B

NikuHBS · Answer

Total number of codes is 9*10 = 90
The following pairs can create confusion
19/61, 16/91 , 18/81, 68/89 , 69/96, 86/98---> Total 12 numbers

Thus, the remaining 78 numbers will create no confusion.
 
Option C

urshi · Answer

Option B : 71

NikuHBS · Answer

Even though i strongly feel that 66/99 can also create confusion, leading to a total of 14 confusing number and therefore that answer should be 76 but there is no such option.

19/61, 16/91 , 18/81,  66/99 , 68/89 , 69/96, 86/98

GMATBusters  can you verify this part once.

saurabh851 · Answer

ANS: C (78)

Best way should be, No confusion codes = total codes - confusion codes
Total possible codes having first non zero digit...
9 * 10 = 90

Confusing codes can only be possible by 4 digits 1,8,6,9
O can be read as same upside down but we have restrictions as first digit should be non zero....so 0 is ruled out (for example 10 read as 01.....or 60 read as 09)
Possible confusing codes, 4 * 3 = 12

Then, Non- confusing codes = 90-12 = 78

Hope it is right...

KG04 · Answer

0,1,6,8,9 => These 5 digits create confusion
No of codes with 2 distinct digits and first digit non zero = 9*9 = 81
Number of digits which create confusion = 4 * 4 = 16 (1,6,8,9 for 1st digit and 4 out of the 0,1,6,8,9 for second digit)
These 16 numbers include 69,96,10,60,80,90 but these do not create confusion hence would not be counted
=> Codes without confusion = 81-16+6 = 71

yashikaaggarwal · Answer

The 2 digit no. Whose tens place is non zero= 10,11,12..........99.
Total No. = 99-10+1 = 90
Since no. Are distinct, therefore= 99-9 = 81
The number when is moved upside down creates confusion = 16,18,19,61,68,81,86,89,91,98 = 10 no.
So total of such 2 digit which don't create confusion= 81-10 = 71

IMO 71

Posted from my mobile device

ashwini2k6jha · Answer

Answer B
The digits which create confusion are 0, 1, 6, 8, 9.
The total number of two-digit codes having distinct digits and first digit non-zero.
 1st place 2nd place 
9 option 9 options ( one digit is already used out of 1 to 9 and 0 is included)
(1 to 9) 
 
 = 9×9 =81 such codes
Total number of two digit codes which can create confusion.
 1st digit 2nd digit 
4 options 4 options
(1, 6, 8 or 9) (one digit is already used out or
 1, 6, 8 and 9 and 0 is
 also included ) 
 =4×4=16
But these 16 two-digit codes include 69 and 96, which create no confusion. Apart from these, 10, 60, 80 and 90 are such two digit codes as create no confusion because these codes are no expected. 
Hence total number of two-digit codes which create no confusion
=81-16+6=71
Answer B

vtexcelgmat · Answer

firas92 · Answer

The digits that can cause confusion when read upside down are 1,6,8 and 9 (We will not consider 0 since we know that a code always starts with a non zero digit)

So the question basically asks us to find the number of codes which are not made up of two digits from 1,6,8 and 9

Total Number of codes that can be created = 9*9 = 81 (9 options for the first digit since it is non zero and 9 options for the second digit including 0)

Number of codes that can be created from the digits 1,6,8 and 9 = 4C2*2! = 12 (Select two of the four digits and the two selected digits can be arranged in 2! ways)

So the total number of codes that cannot create confusion = 81-12 = 69

But among these 69 and 96 read upside down will be tge same

So 69+2=71 codes

Answer is  (B)

MayankSingh · Answer

IMO B

Total Possible Codes = 9 x 9 = 81

Confusing codes:  
 Digits wherein upside down can create confusion = {0, 1, 6, 8, 9}
Note: Since it is know that first digit of the code is nonzero, so cases with upside down reading with first digit 0 isn't possible (eg. 10/60/80/90 -> 01/09/08/06 - reader will know that 0X isn't possible and he misreading)

So possible cases will be from digit = {1,6,8,9}
Total cases = 4x3 = 12
Out of these 2# cases = 69 & 96 will appear the same upside down.
So net confusing cases = 12-2 = 10

Numbers with no confusion = 81 - 10 = 71
 
 Ans. (b) 71

Kinshook · Answer

Given: 
1. An Agency forms a code of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero. 
2. The code, handwritten on a slip, can however potentially create confusion, when read upside down-for example, the code 91 may appear as 16. 
Asked: How many codes are there for which no such confusion can arise?
(a) 69
(b) 71
(c) 78
(d) 80
(e) 89

Total number of codes of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero = 9*9 = 81
Digits causing confusing when read upside down = {1,6,8,9}

Total number of codes creating confusion when read upside down = 4*3 = 12; Excluding the possibility of second digit = 0; since when read upside down it will become invalid.
16,18,19, 61,68, 69 , 81,86,89, 91, 96 ,98 
Out of the 12 confusing codes, 69 & 96 will not create confusion when read upside down.
Total codes which can create confusion = 12 -2 = 10

Number of codes for which no such confusion can arise = 81 - 10 = 71

IMO B

Archit3110 · Answer

total such numbers which can cause confusion ; 1,6,8,9
total possible arrangements ; 9*9 ; 81
so tens place we have 4 and for units place we will again have 4 options as digit is not repeated and 0 is also excluded
4*4 ; 16
also we have is 10,60,80,90 , 69,96 ; 6 digits which cause no confusion 
81-16+ 6 ; 71
OPTION B

An Agency forms a code of two distinct digits selected from 0, 1, 2, …., 9 such that the first digit of the code is nonzero. The code, handwritten on a slip, can however potentially create confusion, when read upside down-for example, the code 91 may appear as 16. How many codes are there for which no such confusion can arise?
(a) 69
(b) 71
(c) 78
(d) 80
(e) 89

GMATWhizTeam · Answer

Solution 
 • The number of codes of two distinct digits such that first digit is non-zero = 9*9 = 81
• Now, confusion will arise if the two-digit code is formed by digits, 1, 6, 8 and 9.
 o So, the number of codes of distinct two digits that can be formed by 1, 6, 8, and 9 = 4*3
o However, 69 and 96 will read same if read upside down. 
• Therefore, the number of required codes which does not create confusion = 81 – 12 + 2 = 69 + 2 = 71

Thus, the correct answer is  Option B.

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