Q.Can we construct a unital commutative C* algebra such that it admits exactly 5 non trivial projection ?
I can't conclude that answer. I only know For some C* algebra , I and O is the only projections. If we take K as a connected compact Hausdorff space , C(K) does not have projection other than the constant functions I and 0 .
Is there any C* algebra satisfying the above question mentioned requirement?
A unital commutative C$^*$-algebra is of the form $C(X)$ with $X$ compact Hausdorff. The projections are precisely the characteristic functions $1_E$. Since you need $1_E$ to be continuous, the set $E$ needs to be clopen, that is an open connected component.
Let $E_1,\ldots,E_m$ be the connected components of $X$. Now let's see what happens:
$m=2$: here $X=E_1\cup E_2$, and the projections are $0,1_{E_1},1_{E_2},1$
$m=3$: now $X =E_1\cup E_2\cup E_3$, and the projections are $0,1_{E_1},1_{E_2},1_{E_3},1_{E_1\cup E_2},1_{E_1\cup E_3},1_{E_2\cup E_3},1$, so the non-trivial projections are already six.
$m>3$: the number of available projections will be greater than six. Actually one can figure out that the number of non-trivial projections is $2^m-2$.
It might be easier to see what's going on if you instead consider $A=\mathbb C^m$.