JEE Main 2022 · 29 July · morning · Numerical answer · +4 / −0
The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.
Answer and solution
Answer 282
Solution
Maths sir · 3 steps
The entry sum is just the number of 1s, so choose which of the 9 cells hold a 1.
- Sum means count. Every entry is 0 or 1, so the sum of all entries equals the number of cells that hold a 1. A 3 × 3 matrix has 9 cells, so the sum is a whole number from 0 to 9.
- Allowed sums. The primes from 0 to 9 are 2, 3, 5 and 7. Note that 1 is not prime. If the sum is k, you choose which k of the 9 cells get a 1, and that can be done in C(9, k) ways.
Sum k (number of 1s) Ways C(9, k) 2 C(9, 2) = 36 3 C(9, 3) = 84 5 C(9, 5) = 126 7 C(9, 7) = 36 36 + 84 + 126 + 36 = 282
- So. There are 282 such matrices.
The common slip: 291. Counting 1 as a prime adds C(9, 1) = 9 extra matrices and gives 291, not 282.
The idea to keep. For a 0-1 matrix, any condition on the sum is a condition on how many 1s there are. Pick the allowed counts k and add up C(n, k) over them.
Answer from the NTA final answer key. Solution by PrepWiser.