JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Matrices

The number of matrices of order

3 \times 3

, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is ______.

Answer: 282

Solution

3 \times 3

matrix has 9 entries, each either 0 or 1. The sum of all entries can range from 0 to 9. The prime numbers in this range are 2, 3, 5, and 7.

The number of matrices with sum equal to a prime is:

Sum = 2: $^{9} C_{2}$ ways to place two 1's
Sum = 3: $^{9} C_{3}$ ways
Sum = 5: $^{9} C_{5}$ ways
Sum = 7: $^{9} C_{7}$ ways

Total =

^{9} C_{2} +^{9} C_{3} +^{9} C_{5} +^{9} C_{7} = 36 + 84 + 126 + 36 = 282

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Integer answer

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