Gary, the main goal is success on climbing a 14er. Thus if you don't end up with permits either way as Bob P mentions, I would strongly recommend White Mt as this would be the easiest way for Barb and Bri to bag a high peak. Climbing White would give them alot of confidence for the next attempt at Whitney and show them that they can handle the altitude OK & besides, the road is alot easier to hike than a rocky trail.

Of course Cottonwood and Langley would be possible too and probably alot more scenic, but White would be easier and not as challenging as Langley or Whitney for that matter.

Sometimes you have to start with lower expectations and you know how you felt after trying Whitney and had to come back the next year to finally reach the summit. Barb and Bri need success on this trip in July !! Still I am betting you get your permit OK and with Joe along everyone will have a good shot at the Whitney summit.