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**Hint:**Half life of a radioactive substance is the time taken by radioactive substance to become half of the initial amount by radioactive decay. Thus calculating the time of change in amount by half gives the half life.

**Complete answer:**

As half life refers to the time required to change the amount of the decaying substance to its half amount. The half life of a substance is very important as it gives the information about the actual amount of substance present in the considered substance.

\[{}^{14}C\] is a radioactive element. The half life of \[{}^{14}C\] is given as \[5760years\]. The initial amount of \[{}^{14}C\] present is \[200mg\]. Thus the time taken for the initial amount of \[200mg\] to become \[100mg\] is \[5760years\]. The radioactive decay of \[{}^{14}C\] is shown as:

\[200mg\xrightarrow[{\dfrac{1}{2}}]{{5760years}}100mg\xrightarrow[{\dfrac{1}{2}}]{{5760years}}50mg\xrightarrow[{\dfrac{1}{2}}]{{5760years}}25mg\].

After the first decay of radioactive \[{}^{14}C\] the amount left is \[100mg\]. Then a second decay occurs for \[5760years\] to give \[50mg\] followed by a third decay for \[5760years\] to give \[25mg\]. Hence three half lives are required for the radioactive \[{}^{14}C\] to change from \[200mg\] to \[25mg\].

Thus the total number of years required by radioactive \[{}^{14}C\] to reduce to \[25mg\] is

= $3 \times 5760years = 17280years.$

**Hence option \[\left( 1 \right)\] is the correct answer.**

**Note:**

Half-lives are characteristic properties of different types of highly unstable nuclei of various elements and the particular way in which they decay. The radioactive decay of alpha and beta particles are slower than the decay of gamma particles. The half-lives for beta decay range are higher than one-hundredth of a second and for alpha decay it ranges higher than about one one-millionth of a second. Half-lives for gamma decay are very short around \[{10^{ - 14}}\] second. Thus isotopes with longer half life decay slowly than the ones with shorter half life.

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