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In my textbook it is written that Tritium is an isotope of hydrogen whose nucleus Triton contains 2 neutrons and 1 proton. Free neutrons decay into p + electron + anti-neutrino . If one of the neutrons in Triton decays, it would transform into $He^3$ nucleus. This does not happen and the reason is given to be that both the neutrons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a $He^3$ nucleus.

But, why is necessary that both the nucleus should decay simultaneously?

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It does. Halflife a bit over 12 years. Who told you it doesn't? – dmckee Mar 3 '14 at 16:19
    
@dmckee Sir can you please explain?I'm not able to understand from the given link. – Rajath Krishna R Mar 3 '14 at 16:22
    
That is from an on-line table of the isotopes (useful link, BTW). The symbol at the top left is $^3_1\mathrm{H}$, which is to say tritium. The table lists the decay mode b- (which means $\beta^-$) as having a 100% branching ratio and no other modes (i.e. all decays occur in that channel) and gives the halflife. For more complicated isotopes there is a lot more data in the page, but tritium is simple. – dmckee Mar 3 '14 at 16:25
    
There is no double beta decay mode listed as it would be energetically disfavored. – dmckee Mar 3 '14 at 16:27

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