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<?php
add_filter('upload_mimes', 'custom_upload_mimes');
function custom_upload_mimes ( $existing_mimes=array() ) {
  // add your extension to the array
  $existing_mimes['swf'] = 'application/x-shockwave-flash';
  $existing_mimes['doc'] = 'application/msword';
  $existing_mimes['docx'] = 'application/vnd.openxmlformats-officedocument.wordprocessingml.document';
  $existing_mimes['pdf'] = 'application/pdf';
  // add as many as you like
  // removing existing file types
  unset( $existing_mimes['exe'] );
  // add as many as you like
  // and return the new full result
  return $existing_mimes;
}