Typically, the permission-denied error appears when attempting to execute the /bin/bash binary when either permissions problems with the root folder or mounting problems with a Live distribution exist. Although other problems do cause the error, they are typically a derivative of one of the two, either dealing with permissions or file-system mounting issues. If you are unable to fix your system using these techniques, try asking in your distribution's IRC channel for guided help.
Type "ls -la /" from the command prompt to view a list of permissions.
Type "chmod 755 /" if the output does not read "drwxr-xr-x", commonly referred to as 755 permissions. Otherwise, continue.
Type "mount | grep sd" to view the mounted hard drive devices. If one reads noexec, then you are unable to execute binaries on that drive and it is likely your root partition.
Type "mount /dev/sda1 -o remount,exec" to remount the partition with the ability to execute binaries. Replace "/dev/sda1" with the drive name that showed the issue before from the mount command.