If the average of 50 numbers N₁, N₂, ........... N₅₀ is M. Then the average of the numbers N₁-100, N₂-100, ........... N₅₀-100 will be:

The correct option is M − 100.

Let's derive the new average step-by-step.

We are given that the average of 50 numbers $N_1,N_2,\dots ,N_{50}$ is $M$.

By definition, the average is the sum of all numbers divided by the total count of numbers:

$\frac{N_1+N_2+\dots +N_{50}}{50}=M$

Multiplying both sides by 50, we get the sum of the original 50 numbers:

$N_1+N_2+\dots +N_{50}=50M$

Now, we need to find the average of the new set of numbers: $N_1-100,N_2-100,\dots ,N_{50}-100$.

The new average is given by:

$\text{New Average}=\frac{(N_1-100)+(N_2-100)+\dots +(N_{50}-100)}{50}$

Grouping the terms together:

$\text{New Average}=\frac{(N_1+N_2+\dots +N_{50})-(100\times 50)}{50}$

Substitute $N_1+N_2+\dots +N_{50}=50M$ into the expression:

$\text{New Average}=\frac{50M-5000}{50}$

$\text{New Average}=\frac{50(M-100)}{50}$

$\text{New Average}=M-100$

Thus, if a constant value (100) is subtracted from each term in a dataset, the average of the dataset decreases by that exact same constant value. The new average is M − 100.