I want to express the number of customers my company has. Is it better to use customer number or customer count?
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Avoid “customer number” because it is likely to be misunderstood. “Customer number” commonly refers to a customer identification number. You will be understood if you say “customer count”. Some other possibilities are “number of customers” and “total number of customers”.
You want a 'count'
Number is too ambiguous in this case, because it is usually something specific to a customer (think PIN - personal identification number) rather than a tally of the customer set.