JEE Main 2026 — Mathematics Probability & Statistics

The given numbers are $21, 8, 17, a, 51, 103, b, 13, 67$21,8,17,a,51,103,b,13,67. There are $9$9 numbers in total.

Given that the mean is $40$40, the sum of the numbers is:

$$\sum x_i = 9 \times 40 = 360$$∑xi​=9×40=360

The sum of the known numbers is:

$$21 + 8 + 17 + 51 + 103 + 13 + 67 = 280$$21+8+17+51+103+13+67=280

Therefore, the sum of $a$a and $b$b is:

$$a + b = 360 - 280 = 80$$a+b=360−280=80

The median of the $9$9 numbers is $21$21. Arranging the known numbers in ascending order, we get:

$$8, 13, 17, 21, 51, 67, 103$$8,13,17,21,51,67,103

There are $3$3 numbers less than $21$21 and $3$3 numbers greater than $21$21. For $21$21 to be the median (the $5^{\text{th}}$5th observation), one of the unknown numbers must be $\le 21$≤21 and the other must be $\ge 21$≥21. Since $a > b$a>b, we must have:

$a \ge 21$a≥21 and $b \le 21$b≤21

The mean deviation about the median is $26$26, so the sum of absolute deviations from the median is:

$$\sum |x_i - 21| = 9 \times 26 = 234$$∑∣xi​−21∣=9×26=234

The sum of absolute deviations of the known numbers from $21$21 is:

$$|8 - 21| + |13 - 21| + |17 - 21| + |21 - 21| + |51 - 21| + |67 - 21| + |103 - 21|$$∣8−21∣+∣13−21∣+∣17−21∣+∣21−21∣+∣51−21∣+∣67−21∣+∣103−21∣

$$= 13 + 8 + 4 + 0 + 30 + 46 + 82 = 183$$=13+8+4+0+30+46+82=183

The sum of absolute deviations of $a$a and $b$b from $21$21 is:

$$|a - 21| + |b - 21| = 234 - 183 = 51$$∣a−21∣+∣b−21∣=234−183=51

Since $a \ge 21$a≥21 and $b \le 21$b≤21, we can remove the absolute values:

$$(a - 21) + (21 - b) = 51$$(a−21)+(21−b)=51

$$a - b = 51$$a−b=51

We now have a system of two equations:

$$a + b = 80$$a+b=80

$$a - b = 51$$a−b=51

Adding the two equations, we get:

$$2a = 131$$2a=131

Answer: $131$131