  We present a new method for determining the sensitivity of X-ray imaging
observations, which correctly accounts for the observational biases that affect
the probability of detecting a source of a given X-ray flux, without the need
to perform a large number of time consuming simulations. We use this new
technique to estimate the X-ray source counts in different spectral bands
(0.5-2, 0.5-10, 2-10 and 5-10keV) by combining deep pencil-beam and shallow
wide-area Chandra observations. The sample has a total of 6295 unique sources
over an area of $\rm 11.8deg^2$ and is the largest used to date to determine
the X-ray number counts. We determine, for the first time, the break flux in
the 5-10 keV band, in the case of a double power-law source count distribution.
We also find an upturn in the 0.5-2keV counts at fluxes below about
6e-17erg/s/cm2. We show that this can be explained by the emergence of normal
star-forming galaxies which dominate the X-ray population at faint fluxes. The
fraction of the diffuse X-ray background resolved into point sources at
different spectral bands is also estimated. It is argued that a single
population of Compton thick AGN cannot be responsible for the entire unresolved
X-ray background in the energy range 2-10keV.
