Radioactivity

Radioactive decay, as NCERT describes, is a random process in which the number of surviving nuclei halves during each half-life $T_{1/2}$. After $n$ half-lives the fraction remaining is ${\left(\frac{1}{2}\right)}^n$, so $N=N_0{\left(\frac{1}{2}\right)}^n$. Starting from $N_0=8$ billion and letting $n=3$, we get $N=8\times \frac{1}{8}=1$ billion nuclei. The value 2 billion is wrong because it applies only two half-lives ($8\to 4\to 2$). The value 4 billion is wrong because it applies a single half-life. The value 0.5 billion is wrong because it applies four half-lives, one too many. As stated in NCERT Class 12, Chapter 13 (Nuclei), decay is exponential and each half-life removes half of whatever remains, independent of the starting amount, because decay is a purely statistical process governed by a fixed probability per nucleus. A plausibility check: three successive halvings of 8 give $8\to 4\to 2\to 1$, so exactly one billion nuclei should survive, matching the exponential rule.