CAT 2021 Slot 3 · Quant · Question 4
CAT 2021 Slot 3 previous year question on Permutation and Combination, with the answer and a step-by-step solution.
A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is
Answer & solution
Answer: 50
Counting directly means splitting into cases; it is far quicker to count everything and subtract.
With each of the 4 places free to be 1, 2 or 3, there are {3}^{4}=81 numbers in all.
Numbers with no 2 use only \{1,3\}: {2}^{4}=16. Numbers with no 3 likewise: {2}^{4}=16.
Those two groups overlap in the numbers that have neither 2 nor 3, that is 1111 alone: 1 number. By inclusion and exclusion, the numbers missing at least one of 2 and 3 number 16+16-1=31.
So the count is 81-31=50.
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