Find the number of 5-digit numbers containing only the digits 1 and 2 (must have

Question

Find the number of 5-digit numbers containing only the digits 1 and 2 (must have a 1 and must have a 2). There are 3 types of donuts at a bakery - types A, B, and C. A number of people during the day came and bought donuts. Some statistics were kept oil buying patterns: 125 people bought type A, 75 people bought type B, 130 people bought type C, 30 people bought type A and B, 35 people bought type B and C, 85 people bought type A and C, 10 people bought all three types. How many people came into the bakery and bought donuts?

Explanation / Answer

total no of five digits number

first let fixed 1 and 2 in last two places, then the remaining 3 places can be filled in 2*2*2 ways. also we can fix 1 and 2 in 2! ways. so total no of this type of digit is 2*2*2*2!= 16. Now again the two digits can be fixed in two different places in 5C2 ways. so total no of five digit numbers is 16*5C2= 16*10= 160.

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