The mean of a set of 30 observations is 75. If each observation is multiplied by a nonzero number $$\lambda$$ and then each of them is decreased by 25, their mean remains the same. The $$\lambda$$ is equal to $$\{0\}$$:

First, we recall the definition of the mean. If a set has $$n$$ observations whose sum is $$\text{Sum}$$, then the mean $$\overline{x}$$ is given by the formula

$$\overline{x}=\dfrac{\text{Sum}}{n}.$$

We are told that the mean of the original 30 observations is $$75$$. Using the definition of mean, we find the total of these observations:

$$\text{Original Sum}=30\times 75=2250.$$

Next, every observation is first multiplied by a non-zero constant $$\lambda$$ and then each of these products is decreased by $$25$$. Hence an individual observation $$x$$ is transformed to

$$y=\lambda x-25.$$

We now compute the sum of all the new observations. First, multiplying every original observation by $$\lambda$$ multiplies the whole sum by $$\lambda$$, so the intermediate sum becomes

$$\lambda\times 2250=2250\lambda.$$

After that, subtracting $$25$$ from each of the 30 observations reduces the total by $$30\times 25=750$$. Therefore the final sum of the transformed observations is

$$\text{New Sum}=2250\lambda-750.$$

We are told that the mean “remains the same,” i.e. the mean of the new observations is also $$75$$. Using the mean formula again with the same number of observations (30), we write

$$\dfrac{2250\lambda-750}{30}=75.$$

Now we simplify step by step. First multiply both sides by $$30$$ to clear the denominator:

$$2250\lambda-750=75\times 30.$$

Since $$75\times 30=2250$$, this becomes

$$2250\lambda-750=2250.$$

Next, add $$750$$ to both sides:

$$2250\lambda=2250+750.$$

Compute the right-hand side:

$$2250\lambda=3000.$$

Finally, divide both sides by $$2250$$ to isolate $$\lambda$$:

$$\lambda=\dfrac{3000}{2250}=\dfrac{3000\div 750}{2250\div 750}=\dfrac{4}{3}.$$

Thus, the constant $$\lambda$$ must be $$\dfrac{4}{3}$$.

Among the given choices this corresponds to Option B.

Hence, the correct answer is Option B.