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Counting Quadruplets (Posted on 2010-08-20) Difficulty: 3 of 5
Each of A, B, C and D is a positive integer with the proviso that A ≤ B ≤ C ≤ D ≤ 20.

Determine the total number of quadruplets (A, B, C, D) such that A*B*C*D is divisible by 50.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 5 of 11 |
50 = 2*5*5

Let P={5,10,15,20}, q = set of remaining less than 21

to be divisible by 50, at least 2 must be from p.

Case 1: 4 chosen from p

No = 1

Case 2: 3 chosen from p and 1 from q
4C3 * 16 = 4*16 = 64

Case 3: 2 from p and 2 from q
if chosen = 5,15
one from remaining 2 must be even.
No = 8*15 = 120
(8 ways even can be chosen and remaining in 15 ways)

if chosen not equal to 5,15.
Total no. of ways 2 chosen p = 4C2 = 6
excluding previous case its 5
product of these 2 will be divisible by 50 itself
No = 5*16C2 = 5*8*15 = 600

Total = 1+64+120+600 = 785

  Posted by Praneeth on 2010-08-20 17:29:13
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