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7mm-08 is what my daughter shoots. Same case as the .308 and .243, larger bullet than the .243 and better BC than the .308.
Their all good cartridges at closer ranges. I would go with the .300 sav,.308 or .270 using the heavest bullets they make. They should do fine out to 200yds. Althow a 11 year old should keep it down to 150 or so. He practess a lot at the range to see what the best grouping is at diffrent distances. Here are some ballistics out of a book called shots at big game by Craig Boddington. Great Book You should pick it up. Lots of info. 25-06 is a little light in bullet weight with 117gr or 120gr=1,668lbs at 200 yards not a good elk round. Good deer round to 300yds.270 win would be best over the 25-06,because you can use 130gr bullets =1,957lbs at a 200yds and 150gr bullets = 1,886lbs at 200yds. Good out to 200 yards for Elk.7mm-08 with 140gr is good out to 100yds =2,142lbs at 100yds A bit light for Elk good deer round to 300yds.300 savage 180gr is good out to 150yds= almost the same,but a little lighter on the foot lbs Good round.308 win with 180gr is good out to 200 yds =1,896lbs at 200yds. Good round The .35 remington using 220gr bullet is ok out to 100 yds for Elk .375 win using the 250gr bullet is good out to 100 yards for Elk"Yes" you can kill with smaller cal. but the .243 win is just to small. Try to keep with heavyer bullets. Like the ones listed below. I would stick with the .300 sav 180gr,.308 win 180gr or 270 win 150gr bullets. You can't go wrong. Look up and read the book listed above or other books about Big game rounds and distances. I find elk ever year that where shot and lost. In most cases the animal was hit good. Elk are the toughest pound for pound. They can take a lot of lead and run for miles. Thats why I try and stick with 180gr .30cal bullets or 175gr to 250gr belt mag loads. I'm sure you will find the right one for your son. I have a .300 savage pump and a .270 win bolt he can shoot. Let me know.