The combined mean of three groups is 12 and the combined mean of first two groups is 3. If the first, second and third groups have 2, 3 and 5 items respectively, then the mean of third group is :

1. 10

2. 12

3. 13

4. 21

Total number of items in three groups = 2 + 3 + 5 = 10

Total sum = Number of observations × Mean

= 10 × 12 = 120

Total number of items in first two groups = 2 + 3 = 5

Sum of first two groups = Number of observations × Mean

= 3 × 5 = 15

The sum of the third group is the difference between the total sum and the sum of the first two groups = 120 - 15 = 105.

By formula,

$\text{Mean} = \dfrac{\text{ Sum of all observations}}{\text{ Number of observations}} \\[1em] = \dfrac{105}{5} \\[1em] = 21.$

Hence, option 4 is the correct option.