Question:

Step 1: Calculate the total sum of all ten observations. Given that the mean of $10$ observations is $15$, the total sum of these observations can be calculated by multiplying the number of observations by their mean. Let the observations be $x_1, x_2, \ldots, x_{10}$. The total sum is: $$ \sum_{i=1}^{10} x_i = \text{Number of observations} \times \text{Mean of all observations} $$ $$ \sum_{i=1}^{10} x_i = 10 \times 15 = 150 $$ Step 2: Calculate the sum of the first six observations. Given that the mean of the first six observations ($x_1, x_2, \ldots, x_6$) is $16$, their sum can be calculated as: $$ \sum_{i=1}^{6} x_i = \text{Number of first six observations} \times \text{Mean of first six observations} $$ $$ \sum_{i=1}^{6} x_i = 6 \times 16 = 96 $$ Step 3: Determine the sum of the observations from the seventh to the tenth. The total sum of all ten observations is the sum of the first six observations plus the sum of the remaining four observations ($x_7, x_8, x_9, x_{10}$). Therefore, we can find the sum of observations $x_7, x_8, x_9, x_{10}$ by subtracting the sum of the first six observations from the total sum of all ten observations: $$ \sum_{i=7}^{10} x_i = \sum_{i=1}^{10} x_i - \sum_{i=1}^{6} x_i $$ $$ \sum_{i=7}^{10} x_i = 150 - 96 = 54 $$ Step 4: Calculate the sum of the last five observations. Given that the mean of the last five observations ($x_6, x_7, x_8, x_9, x_{10}$) is $12$, their sum can be calculated as: $$ \sum_{i=6}^{10} x_i = \text{Number of last five observations} \times \text{Mean of last five observations} $$ $$ \sum_{i=6}^{10} x_i = 5 \times 12 = 60 $$ Step 5: Determine the value of the sixth observation. The sum of the last five observations ($\sum_{i=6}^{10} x_i$) can be expressed as the sum of the sixth observation ($x_6$) and the sum of the observations from the seventh to the tenth ($\sum_{i=7}^{10} x_i$). $$ \sum_{i=6}^{10} x_i = x_6 + \sum_{i=7}^{10} x_i $$ Using the values calculated in Step 3 and Step 4: $$ 60 = x_6 + 54 $$ Solving for $x_6$: $$ x_6 = 60 - 54 $$ $$ x_6 = 6 $$ Thus, the sixth number is $6$. Step 6: State the final answer. The sixth number is $6$. The correct option is Option 1.