$$ \text { If }(\vec{a}+\vec{b}) \perp \vec{b} \text { and }(\vec{a}+2 \vec{b}) \perp \vec{a} \text {, then } $$

Let the population of a species of birds surviving at a time ' $\boldsymbol{t}$ ' be governed by the differential equation $\frac{d p}{d t}-p=-100$. If $p(0)=50$, then $p\left(-\log _e 2\right)$ is equal to

A coffee roaster has $\mathbf{1 2}$ rare coffee beans with intensity scores ranked from $\mathbf{1}$ (mildest) to $\mathbf{1 2}$ (strongest).

You choose 7 beans at random and line them up from mildest to strongest:

$$ C_1< C_2< C_3< C_4< C_5< C_6< C_7 $$

What is the probability that the third bean $\left(C_3\right)$ has an intensity score of exactly 4 ?

A coach needs to select a $\mathbf{4}$-player starting lineup from a pool of $\mathbf{1 0}$ players:

• 5 guards

• 3 forwards

• 2 centres

Find the number of different selections if the 4-player starting lineup must include:

• At least 1 guard

• At most 1 forward

• Exactly 1 centre