PowerShell syntax

What is the usage for first "$" in $($TaskCount)  ?

$smtpBody += "Copy task $($TaskCount): '$($Source)' --> '$($Copy[$Source])'`r`n"
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It's not just the "$" - it's the "$()", and it's called "subexpression".
It allows you to group and evaluate a bunch of commands; in this case, the command is just the value of the variable in question.
It's used mostly where you found it, when a variable inside a string needs to expanded. For simple variables this usually works ("Variable a is $a"), but not when accessing a property of an object ("Property x of a is $a.x") - for the latter to work, you'll need a subexpression: "Property x of a is $($a.x)".
It allows for more complex operations as well:
"Today is $((Get-Date).ToLongDateString())"

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