We perform a quantitative analysis of the decays of $cc\overline{c}\overline{c}$ tetraquarks with $J^{PC}=0^{++},2^{++}$ into 4 muons and into hidden- and open-charm mesons and estimate, for the first time, the fully charmed tetraquark decay width. The calculated cross section upper limit is $\sim 39\left(780\right)$ fb for the 4 muons channel, and $\sim 42\left(616\right)$ pb for the $D^{(⁎)}{\overline{D}}^{(⁎)}\to e\mu$ channel, in the $0^{++}\left(2^{++}\right)$ case. Decay widths depend upon the additional parameter $\xi =\left|{\Psi }_T\right(0){|}^2/|{\Psi }_{J/\psi }\left(0\right){|}^2$, which can be computed with a considerable error. We find $\Gamma \left(0^{++}\right)\sim \Gamma \left(2^{++}\right)=97\pm 30$. On the basis of our results and with the present sensitivity, LHCb should detect both $0^{++}$ and $2^{++}$ fully-charmed tetraquarks.