ssidda01 wrote:GMATGuruNY wrote:goyalsau wrote:Let S be the set of all the two-digit natural numbers with distinct digits. In how many ways can the ordered pair (P, Q) be selected such that P and Q belong to S and have at least one digit in common?
4032
2720
2439
2529
2448
Number of integers in S:
Tens digit can be any digit 1 through 9 = 9 choices.
Units digit can be any digit 0 though 9, excluding the digit used for the tens digit = 9 choices.
Multiplying our choices for each digit, we get:
Total integers in S = 9*9 = 81.
Combinations of P and Q that have at least 1 digit in common = Total ways to combine P and Q - Combinations of P and Q that have no digits in common.
Total ways to combine P and Q:
Number of choices for P = 81.
Number of choices for Q = 81.
Total combinations = 81*81 = 6561.
Combinations of P and Q that have no digits in common:
Tens digit of P can be any digit 1 through 9 = 9 choices.
Tens digit of Q can be any digit 1 through 9, excluding the digit used for the tens digit of P = 8 choices.
Units digit of P can be any digit 0 through 9, excluding the 2 digits already used = 8 choices.
Units digit of Q can be any digit 0 through 9, excluding the 3 digits already used = 7 choices.
Multiplying our choices for each digit, we get:
Combinations with no digits in common = 9*8*8*7 = 4032.
Thus, combinations of P and Q that have at least 1 digit in common = 6561-4032 = 2529.
The correct answer is
D.
Can you please explain why the order of choosing the combinations is Tens digit first for P and Q and then the units digit. If i choose tens and units of P alone i get 9 * 9 ways of getting P. While Q will now have 7 * 7. That doesnt match your answer. Kindly explain?
Start with the MOST RESTRICTED positions.
Neither TENS DIGIT can be 0; the units digits are NOT subject to this restriction.
It is for this reason that I counted FIRST the number of options for each TENS DIGIT.
If we first count the number of options for either units digit, we must recognize a key issue:
Selecting a UNITS DIGIT OF 0 means that we DON'T lose an option for the next tens digit we select, since a tens digit cannot be 0.
Selecting a NONZERO UNITS DIGIT means that we DO lose one option for the next tens digit we select.
Thus, your approach requires that we consider TWO cases:
Case 1: The units digit of P is 0
Units digit of P is 0 = 1 choice.
Tens digit of P can be any digit 1 through 9 = 9 choices.
Tens digit of Q can be any digit 1 through 9, excluding P's tens digit = 8 choices.
Units digit of Q can be any digit 1 through 9, excluding P's tens digit and Q's tens digit = 7 choices.
Multiplying our choices for each digit, we get:
1*9*8*7 = 504.
Case 2: The units digit of P is not 0
Units digit of P can be any digit 1 through 9 = 9 choices.
Tens digit of P can be any digit 1 through 9, excluding the digit already used = 8 choices.
Tens digit of Q can be any digit 1 through 9, excluding the 2 digits already used = 7 choices.
Units digit of Q can be any digit 0 through 9, excluding the 3 digits already used = 7 choices.
Multiplying our choices for each digit, we get:
9*8*7*7 = 3528.
Total options = 504+3528 = 4032.
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