We use high-resolution N-body simulations to study the galaxy-cluster cross-sections and the abundance of giant arcs in the $Λ$CDM model. Clusters are selected from the simulations using the friends-of-friends method, and their cross-sections for forming giant arcs are analyzed. The background sources are assumed to follow a uniform ellipticity distribution from 0 to 0.5 and to have an area identical to a circular source with diameter $1\arcsec$. We find that the optical depth scales as the source redshift approximately as $τ_{1''} = 2.25 \times 10^{-6}/[1+(\zs/3.14)^{-3.42}]$ ($0.63$. The optical depth is sensitive to the source redshift, in qualitative agreement with Wambsganss et al. (2004). However, our overall optical depth appears to be only $∼$ 10% to 70% of those from previous studies. The differences can be mostly explained by different power spectrum normalizations ($σ_8$) used and different ways of determining the $L/W$ ratio. Finite source size and ellipticity have modest effects on the optical depth. We also found that the number of highly magnified (with magnification $|μ|>10$) and ``undistorted'' images (with $L/W10$ and $L/W>10$. We conclude that our predicted rate of giant arcs may be lower than the observed rate, although the precise `discrepancy' is still unclear due to uncertainties both in theory and observations. ; Comment: Revised version after the referee's reports (32 pages,13figures). The paper has been significantly revised with many additions. The new version includes more detailed comparisons with previous studies, including the effects of source size and ellipticity. New discussions about the redshift distribution of lensing clusters and the width of giant arcs have been added