Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X . Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5 . Then the value of 38 p is equal to
Answer: 31
Solution
A five-digit number from these digits is a multiple of 5 iff it ends in 0.
Count numbers ending in 0 (multiples of 5):
Total multiples of 5:
Among these, count those NOT divisible by 20 (i.e., second-last digit is odd):
Total NOT divisible by 20:
Multiples of 20:
, so
Count numbers ending in 0 (multiples of 5):
- First 4 positions use {1,2,2,2}: ways
- First 4 positions use {1,4,2,2}: ways
- First 4 positions use {4,2,2,2}: ways
- First 4 positions use {2,2,4,4}: ways
- First 4 positions use {1,2,4,4}: ways
Total multiples of 5:
Among these, count those NOT divisible by 20 (i.e., second-last digit is odd):
- Pattern _ _ _10: uses {2,2,2} in first 3 positions: way
- Pattern _ _ _10: uses {2,2,4} in first 3 positions: ways
- Pattern _ _ _10: uses {2,4,4} in first 3 positions: ways
Total NOT divisible by 20:
Multiples of 20:
, so
JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Integer answer